Chandlers Ford is nice - in the Hiltingbury (older 60s/70s housing mostly) or top of Valley Park or Knightwood (80s and 90s) areas so you get in the Thorndon catchment - I wouldn't consider Toynbee (other secondary school). The only thing is it is fairly soulless - massive commuter belt basically. I wouldn't describe it as Eastleigh, which is different. Good links to Southampton by car, bus or train. I lived there before they put the train in, and I think they have a night bus to Southampton now too so much better for teenagers than 10 years ago!
We are in Romsey which is fantastic, great schools, close to Southampton and lovely town centre. Less facilities than Winchester but cheaper. Train station, Waitrose, theatre, leisure centre so not that bad!
Southampton - Upper Shirley or Bitterne Park are your only choices really unless you have a girl who you are happy to send to a (to be fair fairly diverse) Catholic school. Upper Shirley High is an improving secondary school in a nice residential area, lovely big Victorian houses, and near the Common, train station etc. Bitterne Park has the best mixed school in Southampton, Victorian and turn of the century housing again with a nice park. But in both cases a few roads makes a big difference. You don't want to be too close to Shirley or the Thornhill housing estate.
Winchester - lovely but expensive. The only bits we could afford are not in the King's catchment anyway (best secondary in the are and probably the most difficult to get into, massive number of applications every year) There is Badger Farm to the south but that is another big commuter estate.
Colleges - Taunton's in Southampton is a mix of vocational and A Levels. If your DC are academic they should try and get into Peter Symonds in Winchester. I went round there recently and I think it is great for teens planning on going to uni. I think they have buses covering most of the south. They take people from as far away as the Falkland Islands so wherever you live you will be OK.
Obviously this is all my completely biased opinion.