Hello!
45 minutes from Hedge End gives you a good spread, although the M27 is a bigger. I know the north/west side of the city better so a bit prejudiced but:
Generally Southampton schools, particularly the secondaries, do not have a good reputation. I've no idea how much of this is historic and how much is truth... People pay £££ to get into the Thornden catchment area, compared with similar properties nearby, but I believe Mountbatten School, Romsey School and The Kings School in Winchester are all good secondaries (although none in exactly cheap areas). Not sure about schools further towards Hedge End.
if you want to be in Southampton itself, there are nicer and not so nicer bits.
Depending on what you can afford, Romsey and Winchester are "naice" towns with associated prices. Chandlers Ford, North Baddesley, Hedge End all a bit surburban although have their charms. I believe Bursledon is cheaper. Netley/Hamble are by the coast but traffic up to the m27 can be a bugger.
Any other desirables?
It's a nice bit of the world. Mixture of coast and countryside, with city thrown into the mix and good transport (trains to London, ferries to France) to get you out too!
I don't know much about them but Botley and Wickham always seem nice places to drive through.