Australia for two weeks

Daniel, it's been a while since I was there, but at the time (and likely still) Quantas had a special "round the country" fare, that let you make a circuit. For example, I did Melbourne-Sydney-Cairns-Alice Springs-Uluru-Melbourne. I would recommend Cairns if you want to get to the Great Barrier Reef. Its a big country, and flying was the only way to travel that much and not waste time.

For a quick trip to Brisbane (or Surfers Paradise and the Gold Coast) look at www.Jetstar.Com, the Qantas low-cost subsidiary.
Lots to do in around Sydney. With a day pass (or longer) from Sydney Ferries you can use the system as a hop on, hop off service to explore the beautiful city. Go to Manly, Bondi, and the Balmain neighborhood, as well as the Rocks and Darling Harbor. Not to mention day trips to the Hunter Valley wine region and the Blue Mtns.
In Melbourne head out for the day onto the Great Ocean Road - one of the world's great scenic drives.
Canberra is a pretty city, but in order of priority for a 1st time visitor I would't rate it near the top. Better to drive out the GOR all the way to Adelaide (another beautiful Aussie city) or even jump on a Jetstar flight for a quick trip to Alice or Uluru.
Smart move doing it as an open jaw. Do lots of research, sort out your particular preferences, and you should be able to put together an enjoyable, compact itinerary that will minimize travel time between cities.

It's a long way to go for only two weeks. I wouldn't bother going for less than three weeks, as by the time you have got over the jetlag it's time to come back.

Distances in Oz are huge, so flying is the best way to get around, but there aren't the cheap airlines such as Easyjet or Ryanair, so it will cost more.

Actually, Jetstar is the Aussie low-cost airline. It's a bit spartan but it is cheap.
One way fares from Melbourne to Uluru (Ayers Rock) are as low as AU$99, or about US$75. Can string together a very cost-effective circuit connecting most of the major cities by planning and booking ahead.

I spent a number of months traveling around Australia on my first trip and would not recommend Canberra or Brisbane if you are taking a short trip of 2 weeks. I would also swap out Cairns and stay around Port Douglas to see the Great Barrier Reef and explore around Daintree National Park. Spending time in the Whitsundays would also be preferable over the Gold Coast.

Depending when you are going next year, I would also suggest spending extra time traveling the length of the Great Ocean Road and go to Kangaroo Island.

Some of your questions are answered here:
https://community.ricksteves.com/travel-forum/beyond-europe/australia-tours

I loved Sydney. 6 days and I hardly saw anything on my list, but was busy and happy the whole time. Specially enjoyed the back-stage tour at the zoo. I took a day tour to the Blue Mountains, but it was during a terribly hot summer and not too scenic, though the tour itself was good.

Melbourne is pretty, not so many sights. The Immigration Museum was extremely interesting and enlightening, had some fun hands-on stuff. Other stand-outs - day tour to Philip Island (fairy penguins w/optional helicopter ride), Queen Victoria market, night-time tour of the Old Melbourne Gaol. I also spent 2 days at the Open - the only slam event with tickets that are easy to buy and affordable, if you're there in January.

I flew between them.

And don't forget Adelaide. We stayed with friends so that may have coloured my pleasure in the place, but we did like it. Canberra was extremely boring and we scrambled away from there quickly.

Thanks so much to all for your time, replies, shared experience and tips. You are certainly helpful and inspiring.
Please, do continue. I am grateful for each and every contribution to this topic.

If you can squeeze another week into your itinerary, that would be ideal. As a 'true blue' Aussie, I hate to put down the nation's capital, however, I think most Aussies would agree, it would not be at the top of their 'to do' list (apart from the National War Memorial). You would need to fly to the Great Barrier Reef. Port Douglas would be my choice and/or the Whitsundays. Not a fan of the Gold Coast.
Jet Star and Tiger Airlines are budget carriers. I know I am biased, but you will not be disappointed if you include Adelaide and South Australia in your itinerary.

I think you could pick two cities, say Sydney and Melbourne and enjoy them in a two week period. I would skip Canberra as others have mentioned. Easy airfares and quick trip to Brisbane. We like Brisbane quite a bit as well as the surrounding areas - particularly the Sunshine Coast area to the north. As others have mentioned, Adelaide is a wonderful city, quite beautiful, easy to get around, and close to some wonderful wine regions.

I think quite a bit depends upon what your interests are and what you want to see and do.

My only trip to Australia was almost 20 years ago. But my single night in Canberra remains memorable. Some of the architecture was very interesting. The Parliament House not only has a green roof long before that was fashionable, but you can walk right up from ground level to the roof, and enter and exit the building either from the lower or roof levels. And many of the embassies are done in exaggerated national fashion (the US Embassy looks like Tara from Gone With The Wind, for instance), so they're fun to see.

But most notable of all was how seeing the capital city brought home just how "young" a country Australia is. Europeans are always struck by how young America is. But Australia is quite a bit younger (they only became a Commonwealth in 1901), and you really see this in Canberra in a way you don't in Sydney or Melbourne.

Again, I was only there one night, and I'm not sure what I would have done had I stayed longer. But don't discount the idea of a short visit, if it interests you.

The other thing I found amusing was that Australians were always dismissive of seeing Canberra, then they admitted they themselves had never been. I'd think they'd want to see what was done with (lots of) their tax money - the Parliament House and the Supreme Court buildings alone cost fortunes!

Yes Australia is very large and so travel between capital cities is done by flying by most. Jetstar as mentioned does occasionally have sales but they also price match and beat it by 10%. I would say any capital could be done in 3 - 4 days but you must consider the amazing sightseeing around the edges and so i would extend that to 1 week minimum.

As others have mentioned .. skip Canberra .. however if you wanted to drive a good old fashioned Road Trip can be loads of fun, then you could fly into Sydney (6 days), drive to Canberra (4 hrs) with a pit stop in Goulbourn for lunch, one full day of Canberra sights, then leave early the next day for Melbourne (7 hrs) with a pit stop in Beechworth for lunch or snack at the Beechworth Bakery. This is 3 days in transit, which could be cut down to 2 days if you set off from Sydney early you could get to Canberra before lunch and spend the afternoon sight seeing and stay one night. Once in Melbourne spend 6 days and fly out from there. any car rental place will let you pick up and drop off in a different city.

if you had an extra week i would spend it in Queensland, a flight in and out would be best

Melbourne is my city and i would recommend 4 days in the city itself, the CBD is very good to just wander around, Hosier lane has great graffiti art, shopping is great, Federation Square/Flinders Street station/St Pauls Cathedral all on one corner there, can get into a fancy restaurant like Heston Bloomenthals during the week, or catch a AFL (Aussie football) match at the MCG Fri-Sunday, general admission tickets always available at the gate. Brighton Beach, Luna Park in St Kilda
And for the surrounds as i was saying, The Great Ocean Road and the 12 Apostles is a good look, go in the other direction down to the Mornington Peninsula and you could easily spend an extra week, Wineries near Red Hill, Peninsula Hot Springs, Cape Schank, and the Moonlit Sanctuary in Pearcedale where you can pat a Kangaroo!

Don't miss the Blue Mountains from Sydney.

Also, the Hunter valley for Winerys.

2 options come to mind.
1) Maybe 6 days in Sydney ( one will be jet lagged) , 3-4 nights in cairns ( as opposed to Brisbane that is just another city albeit a pleasant one), 4-5 nights Melbourne. I quite like Canberra - we live 2 hours from there and go quite often but wouldn't fit it into a 2 week itinerary - maybe if you had two months.

2) just do Sydney and Melbourne and do short trips from there ( eg blue mountains, great ocean road).

What time of year? I think with such a short time period I think flying between stops is the only way to go. Look at all the airlines ( there are only 4) because the "cheap" airlines aren't always the cheapest when you add on checked luggage.