In Australia we're lucky in that all of the mobile phone networks work off the same basic technologies. Broadly speaking, these are known as GSM Networks, and it is the same standard used in most countries around the world.
There are three mobile networks in Australia, those owned and operated by Telstra, Optus and Vodafone. Every other mobile service providers offers connections on one of these networks. There are no exceptions, all providers in Australia offer national coverage.
There are also many similarities between the specific technologies and frequencies that the three networks make use of. This is important because it means that you can use a phone bought from one provider on the networks owned by the other providers.
In short, all of the Australian networks are basically the same and all phones bought in Australia should work on all networks, with all providers.
If you own your own phone, here are a few great BYO phone options to consider: