Australia in two weeks

I highly recommend picking two bases and see what you can from there. We stayed in Cairns and Sydney (10 nights total). From Cairns, we went on a great barrier reef snorkeling trip, saw Daintree Rain Forest and Mossman gorge, Kuranda and the Atherton Tablelands with Millaa Millaa falls. We held a koala at Hartley's Crodcodile adventures. Cairns is a decent size city with more of a youthful vibe. You may consider staying in Port Douglas instead. Either way, rent a car so you can easily get to all of the places you want to go. We stayed at the Hilton on the water in Cairns (we used points so it was free).

We flew between Cairns and Sydney ( about 2 hours). In Sydney, we stayed at the Westin (again on points) within easy walking distance of the harbor. We did not have a car in Sydney. We took a free walking tour our first day so we could get our bearings and see where we wanted to go over the next few days. You can easily take the bus from downtown to Bondi Beach and do the cliff walk. We took the harbor ferry all over, including Manly. We also took a train (public transport) to the Three Sisters in the Blue Mountains. The train takes you into the middle of town but you can catch a bus out to the Three Sisters.

Good advice from Stephen. Would only add a few thoughts:
Consider booking your trip as an open jaw, arriving in Sydney and departing from Melbourne (or the reverse) to keep from wasting a day backtracking to pick up your return flight.
Qantas operates a low cost carrier called Jetstar that provides an economical way to get around the country.
Spending 5 or 6 days in Sydney at the beginning of your journey would allow you to shake off your jet lag while seeing all of the attractions both in the city and in the surrounding area. The Blue Mtns., the Hunter Valley, the beachside community of Manly (everyone's favorite), and Bondi beach are all worth a day and can be accessed via the city's excellent public transport system using an Opal card. Just strolling around Circular Quay and The Rocks on a sunny day is one of life's great pleasures in my opinion. Try to see something at the Opera House if you can manage it.
Great advice already regarding Cairns. Flying in from Sydney, renting a car, and heading up to Port Douglas for several days to see Daintree, the Reef, and the surrounding area is the way to go.
I would try to avoid the temptation to cram too much in during your relatively short stay. The obvious outlier is Ayers Rock. Nice as it is it's just too far out of your way to make sense (IMO). Plus, that time of year it will be plenty hot. I'd give it a miss if it was me.
One of Australia's best kept secrets is the city of Adelaide. Although overshadowed by Sydney and Melbourne, it's a beautiful place in its own right ... lots of inviting parks and green spaces to duck into on a warm day. Collecting a car there would allow you to strike out for Melbourne via the GOR - taking a few days for the journey, just meandering along and enjoying the beautiful views. One place we always enjoy is Tower Hill - an old caldera that's been reserved as an animal sanctuary and which features grey kangaroos, emus, and Koalas in a natural setting with several pretty hiking trails. Can Google it to see if it might be of interest.
You might consider entering Melbourne via the Queenscliff to Sorrento ferry, which will bring you up thru the Mornington peninsula - another very pretty drive. Ditch the car as you enter the suburbs - you really, really don't want to drive in central Melbourne. Their unique rules of the road regarding the trams can confuse native Aussies - don't even think about it.
It's a terrific country, but don't try to see everything on one trip.

you have some very good suggestions so far.
a few things worth considering, that time of year is our hottest, particularly January/feb and the further you go north the hotter.
comparing say melbourne to cairns is like further than say san diego to vancouver.
expect 40+ C days and PLENTY of sunshine , even in Sydney and Melbourne at that time of year.
saying near the coast will be cooler ( if you are very close to the beach) it is generally hot but not so much muggy.

the other big thing to consider is its our XMAS school holidays at that time of year, from roughly 20 Dec till the end of January.
Aussies take this opportunity to have time off with their families and escape the heat by holidaying on the beach at their favourite coastal towns. they book their accomodation 12 months in advance and often go to the same place for decades
I see it ever year at our favourite spot, tourists trying to get a vacancy near the beach, but no luck.

so make sure you get your accomodation sorted! cities like Sydney should be OK, but allow for the huge New years in Sydney ( lots of fire works etc)
https://www.sydney.com/?gclid=EAIaIQobChMI96amnLSd3QIVjxsqCh3nGAHlEAAYASAAEgIRYfD_BwE&gclsrc=aw.ds

try not to see too much, its a huge place with plenty to see, don't ruin your trip by spending it moving around.
hope this helps.

You will love Australia. We returned from a trip to New Zealand and Australia in March. We visited Sydney, Cairns, and Melbourne in Australia. I don’t think I have ever been in a cleaner more beautiful city than Sydney. I never saw any littering or an graffiti in the city. Bathrooms were everywhere and very clean. I would recommend touring the Opera House, taking a harbor tour by boat, and having lunch in the Sydney Tower as a minimum. Cairns was a lot of fun. Watching all the bats, probably 10,000 more was my biggest surprise. They were fruit bats and had wing spans of 4 to 5 feet with some close to 6 feet. I probably talked more about them than the Great Barrier Reef. If you have a chance to fly over the reef in a helicopter, by all means do it. It is only from the air that you realize just how large it is. I would also recommend going to Kuranda for the day. It is only about 1 hour or so from Caines. Be sure to eat lunch at Frogs Restaurant in Kuranda. The Koala Gardens is an ideal place to see Koalas and hold them. The cable car ride over the rain forest was a lot of fun. Unfortunately, the railroad was closed due to flooding, but I have heard great things about it. If you decide to go to Melbourne, be sure to allow a day to go to Phillip Island and see the Penguin Parade at sunset. I really enjoyed the Moonlight Wildlife Conservation Park on the way to Phillip Island. They let you interact with all the animals. I had high expectations for Australia and they were exceeded.

yosemite1 has a good point about Sydney, ( while I don't live in sydney) it is not until you travel around the world do you realise how beautiful Sydney is, it may have ( at times) terrible traffic , but it has a spectacular harbour and surrounds. I have crossed the harbour bridge countless times , but still admire the bridge and the scenery surrounding it.
climbing the bridge is great by the way.
sydney zoo must have the best location of any zoo in the world. easy spot to visit if you want to get your quick 'fix' of native animals. but you can't hold a koala there.
sydney is a good base for day trips, if you have limited time to move around. it has good public transport, particularly the mostly underground trains within the city centre. the airport train takes you into the city is a few stops. just see about an OPAL card.
https://www.opal.com.au/
perhaps- keep an eye out for inexpensive domestic flights by jetstar or virgin australia to get around within Aus.

checkout this entertaining video about sydney.
https://www.youtube.com/watch?v=HdS4DkD33u4
and a suggestion from the video for a zoo visit
https://www.featherdale.com.au/

hope this helps.

We have been to Australia twice and love the country. We didn't go to Ayers Rock, but did see Perth and Darwin on our cruise.
Also, we spent a week in Melbourne and visited the Great Ocean Road as well as Penguin Parade on Philip Island. We did a two day one night tour from Melbourne.

You have a lot of items in your two weeks. You will need 4 nights in Sydney and 4 in Melbourne as well as visiting Cairns for the Reef and Ayers Rock.

The Blue Mountains you do on a day trip from Sydney. Rainforests you can see on the Skyrail from Cairns.

To save $$$ recommend The Tank Stream Hotel in Sydney. Rooms are a bit small, but hotel is in great location. For Melbourne, recommend the small hotel, The Alto on Bourke, located at 636 Bourke Street.

Don't pay over $200US to climb the bridge without your camera, do the piles for less than $20 and take your camera and great photos.