For new arrivals free or heavily discounted medical is slightly complicated. Theres a medicare-surcharge to cover it. Also if your dependants don’t earn enough annually you may get a reduced tax refund as some will be clawed back by ATO. Aus citizens are encouraged to take out private health cover on top. In doing so their levi is reduced.

Some tips:

• The use of bulk-billing facilities can work out to be free
• Dental tourism is becoming popular, you can go to Bali for 7+ days, stay in a 5 star hotel on the beach. While there get treatments. The total cost including treatment, flights, accommodation, food and drink is often less than it would just getting the 1-3 hour treatment in Melbourne, Sydney or Brisbane. 
• canstar.com.au/health-insurance is a helpful comparison site

I hear if you plan to have children in Australia, it pays dividend to have private cover.

I have also heard more than one tale of private health not delivering the right or agreed cover for dental. Often charging for it in peoples 1^st years of policy, then stating it is not effective until 2^nd or 3^rd year when a claim is made.  Read the fine print.