It depends on what your interests are.
I would fly into MEL
and see Melbourne and the Victoria region. There are some nice wineris you could visit. Then fly up to SYD
to do the touristy Sydney stuff (which, retrospectively, is nothing special), then fly to CNS
to see Cairns & Great Barrier Reef. Make sure you go to the Mossman gorge aswell. I was in Cairns in June 2 years ago and the weather was perfect. Then fly to BNE
to see Brisbane & the Gold Coast. While on the Gold Coast go to Surfer's Paradise (again, very touristy), Main Beach (where the Versace hotel is, and Marina Mirage... upscale shopping). In Brisbane, make sure you stay in a CBD hotel, as you have Queen Street Mall on your hands which is Australia's largest shopping strip. You've then got Eagle Street Pier (upscale eating venues), Portside Wharf (same thing), South Bank is a must. Moreton Bay if your keen for something unique.
You can also visit three of Australia's best theme parks (Dreamworld, Wet N Wild & Movieworld) from either Brisbane or the Gold Coast. Distance wise, it's close from the coast, but if your staying in the CBD of Brisbane, you just have to catch a train to Coomera station.
After this, you can fly to SIN
, using either, Singapore Airlines, Etihad, Emirates or Qantas.
If you need anyother help, PM