Going to Australia March 3-15, 2016 - Need Some Help

My first impression is that's a lot of running around given the limited time you have.
In Sydney there's a HoHo bus that will give you a good overview of the city. Here's the website http://www.city-sightseeing.com/tours/australia/sydney.htm.
Try to get out to Manly if you can. It's a beautiful beach community and you can walk out to the North Head if you're really ambitious. Public transport is quite good, especially the ferries, and is an efficient way to get around if you want to revisit somewhere you saw on the HoHo bus.
There are plenty of tour companies that will take you to the Blue Mountains. Can inquire at your hotel on arrival or just find a kiosk in the city - they're everywhere. With only 2 full days in the city you might have a hard time trying to wedge in what is essentially a full day trip.
You can book a GBR tour directly from Cairns, but most would recommend basing out of Port Douglas just to the north, which would require a car. Either way you can research the Trip Advisor "Things to do" section for ideas. Daintree National Park would be my suggestion.
Unless you've already booked the Ghan I'd give it a miss unless you're really (really) into trains and long train journeys. There's not much in the way of scenery to justify a full day working your way thru the "Ghastly Blank" down to Adelaide. It's one of those things that sounds better than it is, IMO. Flying might be a better use of your valuable time.
Adelaide is beautiful - one of Australia's best kept secrets in fact. The Barossa Valley is just north of the city and makes for a very pleasant day trip from the city. From Adelaide the obvious suggestion is to drive the Great Ocean Road back to Melbourne, maybe stopping in at the Tower Hill Reserve along the way for your animal fix if you're interested in seeing mobs of grey kangaroos, emus and the occasional koala along some very pretty bush walks.
At the end of the drive along the GOR you could extend just a bit to take the ferry at Sorrento and then up the Mornington Peninsula into the city. I'd then ditch the car - more trouble than it's worth once your there. Better to rely on public transport. Again, check Trip Advisor to see what might interest you.

You will need skates, wings and perhaps a teleporter to manage the intense schedule you describe.

Sydney - 3 nights
Cairns - 3 nights
Ayers Rock - 1 night
Ghan from Alice Springs - 1 night
Adelaide - 2 nights
Melbourne - 2 nights

Port Douglas is the best location to stay from Cairns. Plan a day to the reef and a day to the Daintree rainforest.

From Adelaide do a day tour to the Barossa for your red wine fix. Melbourne is a place just to hang out.

Sydney options:
Arrival Day - Bondi Beach, and walk to Bronte Beach, Downtown (Town Hall, St Andrews, QVB, Strand Arcade, Martin Place, St James, Hyde Park Barracks, St Mary's)
Day 1 - Botanic Gardens, Opera House, The Rocks, ferry to Manly and return, bridge climb
Day 2 - Blue Mountains - train to Katoomba and cliff walk to Leura via 3 sisters.

The Sydney Gay & Lesbian Mardi Gras Parade is on Sat March 5. Party, party, party !!!

Thanks David and Robert for all of your great suggestions. I think we will be bringing our rollerblades...

Climb on Harbor Bridge is not a free activity.You are very high on the top of the arch, not for those who are afraid of heights, you are clipped for safety and when I was there it cost $150.
I think this post would have been better under Beyond Europe instead of General Europe.

Where are you staying in Sydney? Be aware of street closures on Sat 5th for Mardi Gras. Getting around can be difficult, and I wouldn't want you arriving late for the opera.

I was only in Melbourne and Sydney. What sticks in my mind 7 years later:

Sydney: The Blue Mountains weren't that great - maybe because there was a severe drought that summer and raging wildfires (end of Jan, beginning of Feb), it was about my least favorite thing. I took a "backstage" tour of the Taronga Zoo - really interesting and up close to the animals - but no touching the koalas, even though we were inches from them. Bondi Beach is meaningful for the Aussies but to me, it was just a beach. Lots of fun things to see and do in Sydney.

Melbourne: The night tour of the Old Melbourne Gaol was excellent. So was the afternoon/evening tour to Philip Island to see the fairy penguins. Both the aquarium (SeaLife) and the nearby Immigration Museum were very good.

You need a month. You DO realize how big Australia is? This should be a relaxing vacation.

I am afraid that a month would not be enough. I haven't made it to Blue mountains but can recommend hiking in White mountains where is the tallest mountain of Australia Mt. Kosciuzsko which we conquered.

Lija, it is the Snowy Mountains and I am not sure conquer is the right word. My son strolled to the top of My Kosciuzsko when he was seven. In any event they clearly do not have time to do this given their schedule.

Well, being from Northwest we were little disappointed how easy it was so we added another 40 km and that was not easy.

The Australian continent is the most weathered (some say oldest geologically) on the planet. Hence we have no Alps, Himalayas, Andes or Rockies. The Snowy Mountains are glaciated high plains, and some years the surface area of the Great Dividing Range covered with snow can be larger than Switzerland. However, skiing is not good as we don't have the altitude or reliability. I too walked to the top of Kosciuszko when I was 7 years old. Back then you could drive nearly to the summit in summer, with a 20 minute walk to the top from the car park.
The Blue Mountains may be best left as an option. I think I would lean towards more of Sydney given your limited time. Oh, and you can touch koalas in Brisbane (state laws vary), but alas you have snubbed me !!!

Don't worry djp...we Taswegians have also missed out but we are used to that down here. The rest of the country is always forgetting we exist.

sorry, what's a "Taswegian"?

A person from Glasgow who has migrated to Tasmania.
(But actually without the Scottish. Lol)

David, are the koalas as soft and cuddly as they look?

Soft fur? Yes.
Cuddly? You decide...
http://youtu.be/J6hLIP06I_8

Nigel: http://www.australiandictionary.net/taswegian

And David, you banana benders (or should I say cane toads?) always crack me up :)

Chani. Back in the early 1980s a koala urinated on our national tourism minister as he held it. Not sure if that makes them cuddly but it sure makes them intelligent.

The same minister who had sex with his wife on the ministerial desk and she left her knickers in one of the drawers.

The best thing about visiting Australia is that the Aussies have such a great sense of humor (when you can understand what they're saying :-)

I liked this one better https://www.youtube.com/watch?v=Irrr4ISn5ps

Whilst in Cairns area check out Mosman Gorge and if you have time, a trip up to Daintree is worthwhile.
Good to see you are coming to Adelaide. If it's ' big 'ballsy' red wine you want, you have come to the right place. The Barossa Valley will be the answer to your prayers. There are several Barossa Valley day tours. Like all tours, some are better than others. There are smaller, specialised tours which will take you to the more boutique wineries. which is what I think you will be looking for. I will do some research for you and post some suggestions. Your other day in Adelaide can be spent either taking in the sites of the city on the free hop on hop off bus which does the city loop and also North Adelade. You must take an hour at least so check out the Central Produce Market in Grote Street. If it's a nice day, you can take a 35 minute tram ride down to Glenelg- the closest beach to the CBD. so much to do, so little time!!

Great suggestions. I know we are doing a lot in a short time. We usually do one vacation marathon and one to relax and just stay in one or two places. Went trekking in New Zealand 12 years ago and always said we should have gone to Australia. We are finally doing it. We usually never repeat because they are just too many places to see. This will be a wonderful, exciting whirlwind. Thank you all for your assistance.

Give each day a particular emphasis on the experience at hand ...

Mar 3 - Sydney - Welcome to Oz
Mar 4 - Sydney - Harbour
Mar 5 - Sydney - Beach
Mar 6 - Cairns - Tropical North
Mar 7 - Cairns - Reef
Mar 8 - Cairns - Rainforest
Mar 9 - Uluru (Ayers Rock) - Aboriginal Culture
Mar 10 - Ghan train from Alice Springs - Outback Journey
Mar 11 - Adelaide - City Life
Mar 12 - Adelaide - Country Life (Barossa wine tour ??)
Mar 13 - Melbourne - Multicultural Australia
Mar 14 - Melbourne - Coast Life (Great Ocean Road Tour ??)

I just realised this year you have Adelaide Festival (perhaps the leading arts festival in Australia)
http://www.adelaidefestival.com.au