Hi MrsOz, I have lived in the UK for more than a decade (so my input might not be the most up-to-date) but was born and grew up in Adelaide and most of my family still live there.
Norwood would be a good choice (I used to live there
). It is close to the city centre, but has good local shops, lots of restaurants, and is a nice leafy safe suburb. There is a big shopping centre with supermarkets, butchers, greengrocers, bakers etc (it is common in Australia buy groceries at a supermarket but but meat/fruit&veg/bread from smaller specialist shops). I can't comment much on state schools as i was privately educated but I'd be surprised if Norwood schools were terrible. Adelaide suburbs are pretty homogeneous in terms of residents and Norwood would be mostly a middle-class demographic. There are also lots of private schools in that area, and in general private schooling is much cheaper in Australia than in the UK (about 40% of children are educated privately in Australia compared with 7% in the UK). The school system works a little differently to the UK in that state schools are all secular, and all religious schools (e.g. Catholic schools) are private. Private schools can be religious or secular.
To be honest, you wouldn't go far wrong living in pretty much any of the suburbs close the the city centre on the eastern (e.g. Norwood, Tusmore, Kensington), southern (e.g. Unley) or northern (e.g. North Adelaide, Walkerville) sides. The further you go from the city proper, the "newer" the suburbs get - new builds tend to have less outside space than older houses and can be crammed quite close together on new-build housing estates (acres of soulless rooftops). For this reason I would avoid places like the newer suburbs to the south of the city, even though they are closer to the beaches. Definitely avoid the further-out northern suburbs (Elizabeth etc) which can be, for want of a better word, chavvy.
Adelaide is a very calm city with a slower pace of life. Rundle Mall, the main pedestrian shopping strip in the city, looks half-deserted to me now after London streets, even on a busy day! The southern beaches such as Moana and Sellick's are lovely and not crowded. Closer suburban beaches like Henley beach are also good and have the advantage of nearby beachfront restaurants (avoid Glenelg/West Beach as they tend to be crowded and touristy). The Adelaide Hills also wouldn't be far from Norwood - lots of lovely little towns, a wildlife park, nice scenery. Most suburbs have a large recreation ground with ovals/playgrounds etc, and outdoor public swimming pools (Burnside Pool is not far from Norwood).
Most people drive everywhere (I can't quite get used to driving short distances I would happily walk in the UK when we go back there visiting). Public transport is mostly buses and is OK but not great. Supermarkets have a good range of products, but online supermarket shopping aren't really big there yet, which might be a shock if you're used to the convenience of Ocado/Sainsbury's/Asda deliveries!
The weather gets very hot in summer. In the last few years there have been heatwaves of multiple days of 40C in January/February. When it is this hot you can't really go outside during the day (especially with small DCs). Make sure you house has air-conditioning - it is essential. Winter is pretty mild - temperatures in single figures are rare. Rain is unusual - drought and hosepipe bans are common in summer.
Hope that helps - it's a nice place to live and I hope you enjoy your time there!