The best way to get around Australia is usually to fly.

Buy Qantas, Jetstar, or VirginBlue (sorry, no tie to SQ) tickets online.

Flights to Alice Springs are usually rather expensive. Depending on what your plans are, a package might be the way to go.

Those flight passes are usually much more expensive than buying the individual tickets online.

Cairns-Brisbane is a fairly nice drive if you have the time.

PM me if you have more questions.