There are several phone companies, but the one that was recommended to me by literally everyone was Telstra. Everyone I met before arriving here recommended Telstra, and recommended against Vodafone.
And it just so happened that my family who live here are all on Telstra too so they just added me to their account and I pay my uncle in cash when the money leaves his account. But obviously not everyone can do this.
When we went into the Telstra store they took some details, and they were going to take a $120 deposit from me because I'd only just arrived and didn't have a job etc. We got around that by adding me to my family's account, but without that to fall back on you may need to pay that deposit. It depends on your personal circumstances though.
They offer a SIM only set-up that works like a contract but without the commitment. I pay $25/month for $200 worth of calls and texts and 200GB of data. And that continues until I cancel it, and there's no fee to cancel the arrangement, because it's not a contract.
You can also go for pay-as-you-go, where you top up your money as and when you need, but my cousin said that her friend found the charges were quite high, especially for data, and she was topping up a lot. I've been without data for over four months, so I was quite keen to get the convenience that offers. And with wandering around for various job interviews in every suburb of the city, I've found it very useful.
If you're planning to be in Australia for a long time then a contract might be worth going for, but for me the monthly plan was a good halfway house.