Winchester is good- a bit pricey though! And all 3 comps are very good indeed, esp Kings and Westgate. Bear in mind, though, you'd have to tackle to M3/M27 to get to the Forest from there, bypassing Southampton to its north. The M27 bit from Eastleigh (where the M3 joins) to Southampton is a carpark on workday mornings, heading to So'ton! And of a summer's weekend, from there to Bournemouth is also jammed along the A road that crosses the forest (can't recall its name!).
Winchester has its OWN 'tourists on a summer's weekend' issues! It's also very much London Commuter Belt and the prices (and attitudes??) confirm that!
As for the other places, New Milton is OK, as is Lymington and Christchurch. The Solent 'towns' are Ashurst (still in the forest, really, and pricey), Eling, Marchwood, Dibden, Hythe and Fawley. The closer to the mouth of the Solent one went, the grottier it once was BUT now the oil refineries at Fawley are a lot less messy and smokey in these environmentally conscious times, the whole area is better. Personally I'd be looking Hythe (properly tarted up with a ferry link to central So'ton) and north but perhaps not as far as Totton, unless you're in the Hounsdown catchment part (I THINK Totton is partly 'in'!) Avoid Testwood School. The forest on this eastern side is more heathland, as well but I feel the roads into the forest 'proper' would be less congested that those around Lyndhurst/Brockenhurst.
Brockenhurst College (6th form) is considered excellent.