Most of the quality online casinos you will find are based out of the UK, but do accept players from Australia. 32Red and Platinum Play are two of the top Best Online Casinos UK for Real Money casinos currently accepting Australian players. 32Red is listed on the London Stock Exchange as a publicly traded stock, and gets great reviews from the online casino community.
At this time the Australian govt does not offer online casinos licensees to operate from within Australia, so you will not find any Australian hosted online casinos.