No, there's no National Insurance. There is a Medicare levy, but it's basically included in your income tax overall, and not much. Actually, before I say more, caveat: I'm Australian and DH has PR, so I'm not an expert on the differences when you have a 457 visa - there are certainly different costs/tax, I think. In NSW and other states you will have to pay school fees to state schools if you are on a 457, but not in Victoria (yet!)
Health insurance is complicated, not compulsory, but if you are a high-income earner you will be penalised via taxes if you don't have it. To give you an idea, for us, a family of four, we pay $284 a month for private health, which is hospital + optional extras (dentist, optometry, physio, NOT inc Obstetrics or Gyn etc.) + ambulance. I am about to get rid of it, though, as we just don't really get any value out of it. I had my DS in the public system here, had a wonderful experience, private room with ensuite etc, and didn't pay a thing. For anything serious, you will be treated in a public hospital anyway. For allied health, I'm thinking we might as well just pay out of pocket, but then, none of us are big users. You will still have a co-pay or an excess with private health, so it's really not great value for money, but I would also say that most Australians have it.
For general GPs appointments: either you go to a Bulk Billing clinic and the appointment will be free (charged directly to the government, i.e. Medicare) or you choose a GP which can charge you between $50-70 for an appointment, and you then get $37 back from the government into your bank account overnight. Children will usually be free (bulk billed) at these GPs too. Why des anyone pay to see the GP? Bulk billing clinics tend to have a more NHS feel to them: busier, might rush you a bit, might have more locus, etc, but they are absolutely fine in my experience, especially for minor things. However, if you want a bit more choice over your doctor, you choose to pay the small co-pay. You don't have to register to see a GP, you can go anywhere. I have never had a problem getting a same day appointment here.
Anyway, back to costs. If you own a house, there will be council rates. I can't think of any other day-to-day expenses which are compulsory.
Transport and petrol is generally cheaper than the UK (and trains are a lot cheaper, but not as extensive network.)
Utilities: We live in a 3bed flat. Maximum for gas and electric will be $200 a quarter, probably a lot less.
Schools, I guess: State schools, no fees, maybe a very small voluntary contribution. Catholic Schools: fees, but not that expensive. Private schools: there is a huge range of private schools, at different price points. It used to be the case that the middle class in Melbourne always sent their kids to private schools (even on a modest income), but this is hugely changing, I think. Now that Universities have massive fees (on a debt payback system,) private school fees no longer feel like the price you pay for getting your kids into Uni.
Oh, another thing to keep in mind: Superannuation is a compulsory employer contribution scheme for your retirement. When you see salaries quoted, they are USUALLY + Super of 10%, but it's worth checking if the figure they are quoting is before or after Super. Some employers will also match a voluntary contribution from you, and this will be tax deductible. Some employers (like Universities) will have a higher rate of Super on top of your salary. It's worth clarifying exactly what they are offering. I'm really not sure, if you are thinking of coming short term, how you get your Super back when you leave - I think there must be a way to claim it.
If I think of anything else, I will let you know.