Well dude Its really up to you
I find that when I go on a trip with a limited time frame I ask what am I doing this trip for. Do I wanna get cultural do I wanna drink my face off do I wanna relax in the sun etc.....
I would say if you wanna party your face off drink and get laid then probably most peolpe are gonna say go up adn down the east coast. Gold coast is the region you wanna be byron bay, surfers paradise and everythnig in between. Also good for going out and having a surf and swim and a lounge in the sand!
If you wanna learn about history of Aussie and the coloniztion then I would go from Sydney down to Melbounrne and a few stops in between ther, make stops at such `places as parlement and the old melbourne gaol where they hung ned kelly
And if you wanna get away from it all and really rough it, I would say head to the outback or kimberly range go check out a cattle station and learn about the aboriginal way of life.
If yuo wanna go for the wildlife then probably perth or even better Darwin and that area, less populated as the east coast and still lush and green lots of wildlife to see I think crocidile hunter guy has his farm there.
And if you wanna do the big tourist thing then well tear from Melbourne and the great ocean road all the way up to cairns or brissy do the reef, maybe some skydiving, you can go sailing see the opera house etc....
Again this is just guide what you decide to do is up too you when you get there but Oz is great have fun and cheers mate!