If you go for somewhere in between Portsmouth and Southampton it would give you a broad scope for jobs, there are some large companies in that area anyway (e.g. At whiteley business park & segensworth) but you also have great access to London winchester Portsmouth southsea and Bournemouth.
Fareham is a bit cheaper, you'd easily get something in your price range. Schools I think depends which bit you live in. Town centre bit run down though and traffic not good
Gosport avoid because of traffic
Porchester popular with families, house prices may be a bit more
Whiteley, new, more expensive, nice new shopping centre with restaurants, cinema etc, very family orientated. Houses all new build so bit smaller and less storage etc but easy to maintain and nice area. Has some issues with Broadband so look into that if working from home, some areas are fine but some aren't
Warsash, Locks Heath, Parkgate,Titchfield are great for schools, all are really good primaries and lots of things going on for preschoolers and families. You can get a detatched 4 bed for your price range I think. Great access to motorway etc and it's got shops and whiteley close by. Schools all walkable etc
Knowle village, I haven't been for ages but get the impression it's new build with few amenities so bit of an odd place
Hedge end great for families and schools and has excellent primaries and secondary.