There is Uber in Australia and it is prominent in many of the major cities. If you just want to get around from suburb to suburb, then Uber is a great option. The quality of their service is great and compared to the local taxi companies, they are cheaper to use for short-haul transport.
Melbourne, Sydney and Brisbane have pretty good transportation links within the CBD and the inner suburbs within the city. However, if you want to go further out, you will need to have a car. If you use a taxi or an Uber, the cost to travel will become expensive and at that point, it is actually in your interest to rent or hire a car. Spending $25-50 a day for car hire, which you can split with others in your travelling group is more cost-effective when compared to paying $20-50 for a single journey of 20-40 minutes.
It is definitely the option you will need to consider for long-haul transportation.