Australia and New Zealand are probably two of the safest countries for women who want to travel without any male companionship. There is a thriving backpacker culture in Australia, so big cities and small towns both welcome women backpackers and travellers. This, however, does not apply to all of Oceania, for the traditions and culture in the Pacific Islands are rather different.
Like any other country, Australia has it’s own share of problems. This is the reason why you should exercise basic caution when travelling alone. There are some areas that are considered dodgy and you should try researching on these places so you don’t end up there by mistake. You should certainly not try travelling alone in the Outback or any of the remote areas in Australia, either.
If you would feel more comfortable, you could try researching a list of hotels that are women-friendly. Many backpackers hotels have women-only floors, in case that would make you feel safer. Greyhound buses are a safe mode of transport for those who do not wish to backpack, as well.
As long as you exercise basic caution (as much as you would in your home town), you will be able to have an amazing holiday in Australia.