Australia is a big country, so getting from one city to the next will probably mean that you will need to get on a plane, train or bus.
For a quick trip here are the main airline carriers within Australia:
Qantas (also fly international)
Virgin Australia (also fly international)
If you want to experience the journey a little more, a Greyhound bus will take you anywhere you need to go in Australia.
If you are after something in-between, then have a look at the Rail Australia website for all of Australia's rail adventures.