The National Health Service in Australia is called Medicare. It is funded partly by way of the 1.5% Medicare Levy payable by the majority of income earning individuals, but mostly out of general taxation. Medicare provides free treatment for many ailments in public hospitals, and free/subsidised treatment by doctors and certain dental services. However, many types of treatment are not covered by Medicare - as detailed here.
As with many other free public health services around the globe, demand for "free" Medicare outstrips supply, the result being waiting lists for non-essential services (and often for the necessary ones too). This reality compels many to take out private health insurance so that they have the comfort of knowing that in general they can be treated in a private hospital, maybe in a hotel style room, and when it is convenient in the case of non essential treatments. We have links to some private medical providers here
If you plan for your child to go to a public (State) school you will find your choice will be limited by where you choose to live, particularly at primary school age. The Commonwealth Department of Education, Training and Youth Affairs website has links to the State and Territory Government Education Departments' sites, which in turn will enable you to access websites of individual schools in the area where you intend to live.
If you are planning on sending your child to a private school a good place to start researching is the website of the Association of Independent Schools in the State or Territory in which you will be living. The AIS site will contain links to the websites of individual schools - click here for a list of Associations and links to their sites.
In all cases we suggest that you obtain a prospectus (where available) and talk to the Admissions Tutor of at least three schools so that you (hopefully) have a choice when you arrive in Australia.