All depends on your budget. House prices in Adelaide have gone up in recent years but still low compared to major UK cities. The amazing thing about Adelaide (the thing that a friend reminds me of forcefully whenever I go back there for a visit) is that it is so easy to get around. So if you're living near central Adelaide then you'll be close to both. the sort of suburb you describe is going to be on the more expensive side.
First thing to do is have a look at google maps and type in adelaide, look at the suburbs that neighbour the city centre (or CBD as people say in Australia - stands for central business district)
the suburbs to the north, east and south are the nicer ones and more expensive. The ones to the east will put you in close proximity to the Adelaide Hills. and are still only a 20 min drive to the beach. The ones to the west aren't so green, but closer to beach, cheaper and up and coming.
the more expensive ones are the ones like burnside, st georges, springfield, norwood, kensington, st peters etc. The schools in that area for teens are also good.
Those are all near the city centre. The other thing to think about when considering a move to Adelaide is whether you want to live in the Hills. Will take 30 mins to 40 mins to drive to the beach, but still not that far really. You'll get more bang for your buck and they are stunning in the autumn, near the wineries etc.
Again, take a look at google maps and look at the main road that heads south east out of adelaide. (A1 /M1) Stirling, Aldgate etc are lovely hills towns. and still very close to the city. It would take about 25 mins to drive to the beach from glen osmond for example.
Hope this helps