As I'm sure Tim will agree, with Australia being the large country as it is, you can be limited in such a short space of time (unless you have a healthy budget for air travel)
Personally, I think that if you want to do a bit of everything, then the north east coast (Syd-Cairns) is best, but allow more time up north (Cairns). Darwin is great, and there are some great day\weekend trips, as well as the trip to the rock (allow at least 3 days return by bus, and 2 days at the rock). Worth if if the budget stretches. And don't underestimate the climb to the top if you choose to climb - it's not for the faint hearted...
In relation to trips by backpacker hostels, check it out when you get there, and go with your gut feeling - particularly in Cairns, there are plenty of options!
If you do get to Cairns, a night in the Daintree in the wet season is unforgettable, the wild-life is amazing.
Hope i've been of some help!
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