I moved from the UK to melbourne a few years ago and so all of the advice below applies to melbourne.
Australians love private schools and private health so there are some very average gov secondary schools in wealthy areas (eg: hawthorn high) where all the middle class kids go private. Whereas the inner north (carlton, parkville, northcote, fitzroy) have great public high schools (princes hill, university high, fitzroy HS, northcote HS) and kids from affluent, educated, left wing families (maybe more like islington). I think the same applies to the suburbs in the inner south. There are also some excellent high schools out east but for different demographic reasons.
The better schools website and the myschool website will help you compare. If you like kensington, wimbledon and richmond suburbs like richmond, carlton, albert park, fairfield, south melbourne, elwood, northcote, fitzroy, parkville, camberwell, and williamstown might suit and they all have great public transport and decent or excellent high schools. I think you can find good gov primary schools in most suburbs. Usually, the best schools are tightly zoned and over full as they take everyone in their zone.
The good news is that there is a significant disparity between rents and mortgages so if you are planning to rent there will be few suburbs outside of your budget on that income. Buying would be different.
I think all the suburbs ive listed above have great high streets so you could walk to a chemist, train, butcher etc within 10 mins.
Melbourne does great high streets full of independent shops, cafes and boutiques with few chain shops. The down side is the supermarkets seem small and limited compared to the UK.
When i first came here eveyone kept telling me how 'european' melbourne was (it has a strong but cliched identity). And i kept thinking - which bit of europe? Have you ever been to europe? (They all have) but now ive been here a while and seen a bit more of the rest of australia, i kind of see what they mean.
Australians seem to view not having a licence as akin to choosing to have a disability. They just dont get it. I managed fine without one for a couple of years, living in an inner suburb, but it is definitely great to be able to drive. Its also much easier and cheaper to drive in melbourne than london.
Finally, the weather is different in syd and melb. Syd has a longer summer and swimming season and a milder, shorter winter. I think more rain actually falls in sydney but they have more tropical style downpours.
Australians love swimming so you will find swim squads everywhere. Your kids might like little nippers or surf life saving which might be better in syd?
Renting is more expensive in syd, but if you can afford to live somewhere you like, id go there if you know you love it and you can find a good school. Otherwise, melbourne is always winning 'the most livable city' competitions and deservedly so IMO.