This might be a bit late but I thought I'd wade in anyway. I've lived in this area all my life. My parents live(d) in Gosport, so I grew up there, and I now live in Southsea.
To be brutally honest, I would avoid Gosport like the plague. Especially Forton, Elson, Rowner and Gosport town, exactly as Sidge said. Frankly, it's because of the violence I/friends all witnessed when we were teenagers. I don't want to scare you, but I'd seriously consider if you can afford not to live there (Gosport is officially the cheapest place to buy a house on the south coast- apparently). Don't get me wrong, there's been no shootings, to my knowledge, and young kids are pretty safe, but it all seems to get a bit scary when they grow up.
It's also almost completely devoid of anything to do, and although there's a ferry/night ferry to get you over to Portsmouth proper, I find it rather a palaver.
Southsea, on the other hand, I can't recommend highly enough. Much safer, nicer 'feel', way, way, more to do, easier access to other parts of the city for work/socialising. Not too sure about senior schools, as most are private, but the one that's not is not bad at all. Plenty of nice primaries. BUT, the traffic can get really bad. I don't drive, so perhaps that's why I love Southsea so much. Having said that, Gosport's traffic is possibly even worse, it's not an island, like Pompey, but it's a peninsula, with one, yes one, road out to the big bad world.
I realise that's only two of your possibilities, but hopefully this is useful for them.