It's a complicated question. I've typed out a response 3 times and it ends up being a whole page long and very complicated.
I'll try again and see if I can condense it.
No, healthcare is not free. We have a system called Medicare which everyone is entitled to and is paid for through taxes. If you go to a doctor, they charge you about $70, you pay it and then put in a claim to Medicare. Medicare gives you about $35 back. (There are lots of different ways it works, some people don't pay at all, some people get more back etc but this is the basic way it works).
On top of that, most people also have private health insurance which costs around $5000 per year, depending on what extras you want to be covered for. Private covers you if you need more than just a basic consultation, like if you need surgery etc. We do have public hospitals that are mostly free but waiting lists can be a problem.
If you don't take out private health insurance, the government charges you more for Medicare in your taxes.
What most people tend to do is use public if they need to go to Emergency. You have to wait but they will fix you up and it's free. However, if you then need to be admitted to hospital, you switch to private because you will get quicker and better treatment, can choose your own doctor and have your own room etc. You get wine with dinner
I was again MIA for a long while, I just thought of you guys from Australia and that stuff is pretty damn scary. It's terrifying and infuriating. You go on living your life and you can't predict anything. Some asshole can wreck everything you build in your life. I'm fuming mad just at the thought of it. 
