Religious schools aren't government run like they are in the U.K., so you'd have to pay (although as non-citizens you may have to pay a fee to govt schools as well). Religious school fees are fairly modest though. Don't stress too much about getting a spot, it's much easier than here. As PP said, if you go public your local has to take you. It will most likely be a MUCH bigger school community than you are used to. Average size for primary is 5-800 students. Some public schools have up to 1500 students and 10 classes per year level.
Rent in the areas you're talking about is expensive, as that's a upper middle class area. $200k is generally a good family income but won't go far in that part of town! It's definitely a Mumsnet naice area.
I'm from Melbourne, currently living in London, and will definitely be going back. We lived in the west, which is definitely more working class. Same as here, inner south east is most desirable!
You will be facing a massive cultural adjustment, it was much bigger than I ever expected. However going into jobs for both you and your partner means you will have a ready-made social network (or the start of one) through work which helps a lot.
Budget on needing a car. Public transport is nothing like here.
I personally don't consider Victorian beaches all that, but I grew up much further north where it was actually warm enough to swim! But comparing to the UK you will probably think Melb water is like a bath.
I would definitely say go. Even writing this post has made me miss home a little.