Since all the cities are on the coast, doing 'the four corners and the middle' makes a fair bit of sense.
I'd suggest starting from Melbourne, Victoria (my home town) and going to Hobart, Tasmania; Sydney NSW; Brisbane and then Cairns, Queensland; Darwin, Northern Territory; Learmonth and then Perth, Western Australia.
Then cut inland to Alice Springs (or Ayers Rock Airport if you're using something small or medium-sized); then Adelaide, South Australia, and back to Melbourne.
Allow plenty of time; the various legs average about 800 miles each.
You can fly past Ayers Rock, which shows up well on FS. Look it up on the net to get the direction from Alice or Ayers Rock Field.
To whet your appetite, this is climbing out of Cairns:-
