Coober Pedy, Ayres Rock and Kangaroo Island

You're right in that you really do need to prioritize to make this work. Distances in Australia are vast and travel days can eat up much of your available time.
I'd start by disregarding Coober Pedy and Kangaroo Island...interesting places but just too hard to get to on an abbreviated itinerary. A few days each in Sydney and Melbourne will allow you to see most of the highlights in each city and still have time for a couple of side trips.

My recommendation would be to arrange your trip as an open jaw, flying into Sydney and out of Brisbane, with enroute stops (using Jetstar) to Melbourne, Uluru, and Port Douglas for the GBR. If you're especially ambitious you could add Alice Springs for a taste of the Outback, but know that it's a 300 mile drive from Uluru. Might be worth the one-way drive to see the wild camels that roam in the area.
For a first trip to Australia that would cover the highlights without overdoing it on travel days.
A good resource for off the beaten path driving adventures is "Back Roads Australia" from DK publishing. Can find it on Amazon.
In Sydney, suggest staying in "The Rocks" or, even better, in the beach community of Manly. Get a 3 day pass from Sydney Ferries and use those as a hop on, hop off, service to explore what is to me the most beautiful city on earth.

I'd scratch Coober Pedy for a first trip unless all 3 of you are really interested in learning about opals, opal mining or have a strong desire to experience a stay in an underground hotel with zero sunlight (hope no one is claustrophobic in your family!). If all you want to do is buy yourself some opal jewelry, you will have tons of opportunities to do that in other places in Australia. It's everywhere!

To be perfectly honest, I found Ayers Rock (Uluru) to be sort of underwhelming. If you have spent any time at all in the American Southwest, seeing this lone giant rock, and even visiting nearby Kata Tjuta, just does not seem quite as spectacular as what we have in our own backyard. A lot of tourists choose to climb Uluru, but this is seen by the Aboriginal population as extremely disrespectful. The best time to see the rock is at sunrise (if you aren't early birds, standing around the cold dark desert waiting for the sun to rise might not be everyone's cup of tea), or sunset.

Also, when you stay overnight in that area, you are basically at the mercy of the corporation that pretty much controls the pricing of all lodging and food. Unless you go ultra, ultra luxe, it's sort of like being trapped at Disney, albeit with far fewer choices. The lodging is just OK, the food extremely mediocre and super overpriced.

I'd skip it and use the time to go to Melbourne and travel the full length of the Great Ocean Road heading up towards Adelaide and then I would suggest time spent on Kangaroo Island which was great. I'd choose that over spending any time in Brisbane. The Australian Zoo was ok for a few hours, and it's great that you have a contact who can give you some behind the scenes time, but it's still a zoo. I find it a lot nicer to see the animals in their natural habitat instead of behind bars. If you just want to hold a koala, you could go to someplace like Featherdale Wildlife Park just outside of Sydney and combine that with a day trip to the Blue Mountains.

I agree with basing in Port Douglas if you are going up to the Great Barrier Reef. Much nicer than Cairns. For Sydney, The Rocks is a good (albeit touristy) location to stay as you are within easy walking distance to the Harbour for the Opera House, Bridge, Botanical Gardens and ferries heading towards other points, including Manly - nice beach, fun and very scenic ferry ride, but I don't know that I would want to stay in Manly and spend that much time on the ferries commuting back and forth between Manly and Sydney. "Old" is sort of a relative term in Australia. Unless you are talking about aboriginal areas, it's still a pretty "new" country - like the US. You might see some quaint victorian type architecture scattered about in certain areas, but it's more 1800s/early 1900s you will see - not like Europe.

If you have not purchased guides yet, check your local library and see which ones you like before you purchase.

Another way to string together a possible itinerary would be to fly from Sydney to Cairns first : there are a number of daily excursions out to the GBR directly from Cairns, and the nearby aerial tramway in Kuranda would also be worth a day of your time.
From Cairns you could then fly directly to Adelaide. It's is a beautiful city in its own right and, using it as a base, you could take the ferry out to Kangaroo Island if you really have your heart set on that. You could then drive the GOR one-way back to Melbourne and end your trip with a few days there.
Flying back to the US from Melbourne (open jaw) would preclude having to backtrack thru Sydney and thus wasting a day.
I do think that a side trip to Uluru (and maybe Alice) would be worth it, but you'd need to consult the Jetstar website to see if there are any workable connections to it from Cairns, Adelaide, or Melbourne. It's true that accommodations there are quite expensive - one company has a monopoly on the concession so they really jack up their prices. There is one budget place (a hostel really) called the Pioneer Lodge that tends to sell out quickly so you'd need to book something months in advance. Can check out their website to see if it might be of any interest.

Thanks for the replies so far. I did end up buying the DK Australia book for starters, and am looking at some apps for the iPad. I am concerned that the weather will be a bit cooler than I thought and the daylight hours will be quite limited at the time we usually travel. I am going to have to rethink the Great Barrier Reef since it may be a bit cool for that...

Yes, it may be too cool for the Great Barrier Reef. My husband took his daughters in late July a few years ago and it was very cold, and the water was quite rough. Not quite what they hoped for after going all that way.

Do check the historical weather averages for Cairns before you finalize your decision. August is early Spring there and, although that can result in some seasonal variation, the average daily high is around 80 deg. It will be the middle of "the dry" at that time of year, with relatively low humidity, making it a delightful time to visit.
As I recall the water temp doesn't vary too much out on the reef. Stays pretty temperate all year round, and the snorkeling is uniformly sensational.

We did basically the same trip you are planning with some variations a few years ago, although it was not our first trip, but this one was with two middle-school aged children. I agree with Robert about flying into Sydney, spending a few days there, then flying to Cairns, continue with a flight to Adelaide and drive the GOR to Melbourne and return from Melbourne. It makes the most of your limited time while covering some of the highlights you want.

Regarding weather, we also made the trip in August, and were able to enjoy the GBR and other outdoor activities around Cairns, and while it wasn't hot, it was pleasant. Melbourne was wet and dreary in August when we were there.

We've also considered Adelaide as one of our favorite Australian cities and the access to great wine regions (might not be of interest to you). And we've gone to Kangaroo Island (on a different trip), and while we enjoyed it, I probably wouldn't go back nor would I include it on a "first trip to Australia". To really see the island, IMO you need to have a rental car and stay at least two nights, and the ferry to and from takes up a good part of two days. I know people will do it as a day trip, but I don't think you can really appreciate the island in that short time period.

And plan this trip as you would one to Europe - you'll go back!

This is the company we booked with for our visit a few years ago: http://www.quicksilver-cruises.com/index.php
They run a first class operation. Note that the trips out to Agincourt Reef depart from Port Douglas just north of Cairns. They offer bus transfers if you're staying in Cairns but the easier way would be to just drive up that morning.
Note too that the boat is a large catamaran - makes for a very stable ride for the 90 minute journey out to the reef...even in rough sea states.
And before you have a heart attack while viewing the fares, note that the quotes are in Aussie dollars...not US.

Good advice to base yourself in Port Douglas for your Great Barrier Reef experience. Also recommend Quicksilver Tours for your day on the reef. Flying to Port Douglas then from Brisbane to Adelaide/Adelaide to Melbourne via the Great Ocean Road would work well.
Coober Pedy I haven't seen, but I have been to KI a couple of times, and it would be my pick if I had to choose one destination. If you can squeeze a couple of days on the island that would work well. I love it for it's unspoilt natural beauty. Plenty of koalas to cuddle on the island too.
August in South Australia is 10 times warmer than Scotland and the UK when we were there last Spring!
Keep us posted on your final itinerary

Thank you again for the continued posts. I have started looking at air fares and open jaw flights. Makes the most financial sense to fly into Sydney and out of anywhere else... Any incoming flights to other cities always have a 2 hour layover in Sydney for some reason!

I really like the itinerary from Mother Duck so far.

We have begun laying out a map and putting post-it notes on the places we want to see the most. I've been pricing flights on Jetstar.com between the major cities. Looking to see if the big cities have the "hop-on, hop off" city tour busses that we like when we are new to a city, since they help for orientation.

Our daughter is standing firm on the GBR. Looks like Coober Pedy will be sacrificed due to the fact that it is 8 hours drive to the middle of nowhere and we would have to backtrack on the same road coming back instead of seeing anything else (not the cheapest flight to and from Adelaide either). Their little airport didn't list any other flights except Adelaide, so we couldn't bite the bullet and go someplace else from there.

Also need to think about our rental car... turning it in and renting new ones with any intercontinental flights sounds like a hassle but probably not, I guess.

I am very flexible and easy to please, as is my husband. I am most concerned about keeping our 20 year old daughter happy, (since she usually doesn't like our trips unless they are to London). She doesn't like museums and isn't thrilled about garden tours. None of us scuba dive and we don't snorkel well, so the GBR will mostly be a glass bottom boat tour of some sort. I know she will enjoy the animal sanctuaries or zoos we end up visiting (and she did sound interested in staying underground in Coober Pedy, so she may have some say in that).

I am making a list of everyone's suggestions and will run them by the family. Thanks again!

I logged on to Travelocity and checked non-stop to Brisbane... It came back with zero non-stops from Detroit. Everything on our end goes through someplace in California for layovers. Is the non-stop that you mentioned available on a partner airline not found on Travelocity or Orbitz?

My husband is his own boss, so he wants to limit downtime to 2-2.5 weeks at the most. He originally thought this would be a trip we would take after retirement, but has since changed his mind. Our daughter really wants to do the reef, and the cruise catamaran on http://www.quicksilver-cruises.com/index.php looks like a terrific option for us. I have looked at quite a few photos from some of the blogs on driving (which we are not opposed to)... a lot of the things I am seeing (beaches, hiking trails, dunes, etc.) can be found here in our wonderful state of Michigan! That's why I want to see the koalas and native wildlife so much.

I have tried looking up "beautiful small towns/villages" as alternatives to the big cities. Port Fairy and Port Douglas keep coming up in the searches. Still working things out ~

Sounds like your planning is starting to come together.
Both Port Douglas and Port Ferry would be excellent choices.
The other attraction in Queensland that shouldn't be missed is Daintree National Park. Google it to see if it might be of interest to you and your family.
Another attraction just off the GOR that we highly recommend is Tower Hill - a game reserve in an ancient volcanic caldera that has kangaroos, emus, and koalas running around in a spectacular natural environment. There are a number of easy bush walks that range from short strolls to longer hikes of a few miles. Beautiful place - It's a short drive from Port Ferry near the little town of Koroit. My wife and I thought it one of the most enchanting places we visited during a 3 month trip to Australia a few years ago.

Like the previous poster, I have also stayed in both Port Fairy and Port Douglas as part of 3 months traveling around Australia. I don't know that I would call Port Ferry the town "beautiful", but it's a good stopping point for 1 night if you are wanting to slowly travel the length of the GOR over the course of a few days. It was a great place for bird watching, but I was there in the late summer and you will be traveling in winter so you'd have to do some research on the migration and local wildlife.

Port Douglas had elements of it that were very charming, and other parts that were a little touristy, but it is infinitely better than Cairns. If you are spending some time in the Port Douglas area, Daintree is definitely worthwhile, as is Cape Tribulation. I did go out on the reef with Quicksilver but they really pack a lot of people onto their boats. I much preferred going out by boat from Port Douglas in a small group of just 5 to go snorkeling around the Low Isles. Unlike another poster, I actually found the water to be very clear in the Low Isles and it was excellent snorkeling - but I suppose visibility can be impacted anywhere along the reef by any number of things or the seasons.

You're better off looking at flights from Detroit into Sydney and out of Melbourne (or the reverse). Brisbane is a pretty city and worth a look on a possible future visit, but for a 1st time visitor I wouldn't rank it in the top 10. Connecting via LAX or SFO would likely be your best and cheapest option.
The general consensus on this forum seems to be to broaden your search using one of the popular search engines (Travelocity, Google Flights, etc.), but then to book directly with the airline once you've made your choice - saves a lot of aggravation should there be any issues with the booking later.
Melbourne is worth a few days: the drive around the Mornington peninsula is quite pretty, as is the drive out to Phillip Island with its large colony of Fairy Penguins. Either would make for a pleasant day out with a car, as would a day trip out to Ballarat and the Gold Fields or to the Grampians just west of the city.

Thank you for all the recent replies.

I am thinking of doing a hop-on/off bus tour and staying in The Rocks area in Sydney when we first arrive. A friend has advised us to visit the Sydney tower, and all over my facebook account are ads for viator.com with ideas for tours while we are there... thinking of doing the 2 hour behind the scenes tour of the Taronga Zoo there to get my koala fix in early.

Any ideas on where to stay in The Rocks, or any comments regarding the Taronga Zoo?

We stayed at the Rendezvous Stafford several times on our own travels and liked it very much: http://www.booking.com/hotel/au/rendezvous-stafford-at-the-rocks.html
The apartments are spacious and comfortable, and having a kitchen was a significant money saver for us. Rather than having to dine out for every meal you can be selective regarding when and where you want to splurge a little. Nice location too - it's on a quiet side street and a short stroll to Circular Quay. The Opera House and Botanical Gardens are about 15 minutes further on. The train from the airport takes about 20 min. to arrive at the station at Circular Quay (adjacent to the ferry terminals) and from there it's a short walk of a few blocks to the hotel. Very convenient.
The link for the HoHo buses is here: http://www.city-sightseeing.com/tours/australia/sydney.htm. Note that there are two separate routes. The pick-up point is only a couple of blocks from the Stafford.

I stayed at Quay West Suites Sydney in The Rocks. It was money well spent to get a room facing Sydney Harbor as you could see all the iconic landmarks (Opera House, Bridge - could even see the groups headed up the side of the Bridge for their climbs - highly recommend the climb if you are not afraid of heights) from the enormous windows. Also had a nice view of the cruise ships, ferries and other boats zipping around in the water. One night we unexpectedly got to see a tremendous fireworks display over the Harbor right from our living room. As I had met up with some friends, we had a 2 bedroom suite with separate living room, kitchen and a small laundry room with full-size washer/dryer. Hotel has a pool.

On a return trip when I was on my own, I did not stay in The Rocks, but stayed near Hyde Park at a Travelodge which was actually a very nice hotel in a great location (despite the lack of a view overlooking the Harbor) and it was also much more in the budget range.

The Rocks is one the oldest areas of Sydney - obviously lots of history and character. Taronga Park Zoo is situated over looking Sydney Habour. which adds to its attraction. I'm not a big fan of zoos, but as zoos go, when we visited Taronga many years ago when our now very grown up kids were little, they really enjoyed it. Had several exhibits/animals that our kids had not seen at our local zoo in Adelaide.

Thanks again to the people who have kept posting replies to my forum post. We are in the process of laying out the big map, and it is covered in sticky notes now as we are deciding what will make the final cut. Have been looking at the various hop-on, hop off buses which we will take in Sydney and Melbourne but didn't see one available in Adelaide... Also found a Philip Island bus tour from Melbourne that stops at a nature center on the way down to see the penguin parade. It was advertised on www.viator.com and sounds like a fun day. Advice and comments are always appreciated!

Linda, for what its worth , as others have mentioned, August is our winter and while not as cold as most people experience in the northern hemisphere, be prepared to experience day where the scenery lacks a bit of 'colour'. Although there can be some beautiful winter days ( particularly up north in Queensland)

There is a lot to see within a couple of hours drive from Sydney. so if time is limited maybe its worth investigating some day trips out of Sydney. either north to the vineyards in the Hunter Valley. ( west of Newcastle) Or dolphin/whale watching at Port Stephens, also large sand dunes and beautiful beaches ( just North of Newcastle) these areas are popular with tourists due to their proximity to Sydney. Most places, including the Port Stephens area, have some sort of 'zoo' that allows you to get close to koalas etc.
http://www.portstephens.org.au/

Or a trip to the Blue Mountains/Jenolan caves. west of Sydney. you don't have to travel huge distances before you are into countryside that is 'typical' of the Australian east coast. Personally I would not drive from Sydney to Brisbane, unless you like driving on a highway looking at trees. as most of the nice spots are off the main highway and would take time to visit.
As for seeing animals in the wild, even on the intercity train between Newcastle and Sydney you can generally see kangaroos on the nearby golf course, so while we don't have them jumping down the street ( like some might think ) you don't have to travel to far off areas that can take up time.
As others have mentioned, perhaps stay in the Rocks area in Sydney.

I hope this helps.