Another one in Sydney, so things might be different in NSW than Victoria.
SO my first question is what visa will you be on, as there are differences when it comes to healthcare if you are on a temporary or permanent visa.
There is a public system but it doesn't cover all the cost. A neurosurgeon visit will cost you 320$ and Medicare will pay back 70, so you are out of pocket of 250$. Imaging is hugely expensive and again medicare will only partially cover the expense of an MRi.
Private health will NOT cover for specialist visit or test, but will cover in the private what Medicare would cover in the public, so if you have a surgery with a private surgeon, the private health provider will reimburse you the rebate for that surgery. You will probably be out of pocket in the orders of thousands and your private health will reimburse you hundreds only.
The cost of your stay at the private hospital, not the surgery, will be covered partially or fully by the private fund.
Even if you go through the public system, you can end up with a huge medical bill. My geneticist at public hospital is free, but she sends me to a private neurologist ($640) who then sends me to a private neurosurgeon (300 something) who send me to an MRI $800. Unless you are an in-patient, and seen within the hospital, specialist and test will cost you enormous amount of money even with a platinum top private cover.
Dentist are incredibly expensive and this is when the private funds helps you.
You don't register at a GP, you just go where you want. Some are bulk-billing, which means you don't pay any gap, but a lot pf practices are private so you pay around 75-80$ and medicare gives you back 37, again half out of pocket.
Search for a bulk-billing doctor. But google the reviews on google-reviews (not whitecoat or similar) as they keep both reviews, positive and negative whereas whitecoat will edit response.
In NSW, public school are free for permanent but with a 5000$ fee for temporary resident. Not sure about Victoria. Then of course, you add uniforms and other school fees.
Car: so so so so expensive. You need a double insurance which after 4 years, I still cannot understand. Your personal insurance and the CTP insurance linked to the car and on top the registration. Cars are expensive to buy and run, but without a car, you are lost. Distances are huge, public transport crap. If your son does soccer for example you will have to go to fields in the weekend which can be at 20 min drive.
Early childcare is very expensive, so you need to look at your options for your 3yold.
Time to wake up the kids for school, I have to go. Oh, one last thing for now. Australia is far colder than you think and they don't have heating in the houses, only air con heating or if you are lucky you have a gas bayonet and you can then add a gas heater that you guess cost a fortune $ 1200 for 1 heater!