Hi,
I've lived in Hampton/Hampton Hill area for the last 15 years and it's a really nice place to be when you have young kids. On the positive side all of the state primaries are good and there are also a lot of good independent options if that's the way you want to go.
The local secondary in Hampton is as you say not great and hasn't been great for as long as I've been around here. I think that's partly because a lot of parents in the area choose to go private at secondary or send girls to Waldergrave.
However there is a new school called Turing House which is closer to Hampton Hill just near Teddington Walk in Centre which only opened a couple of years ago. My elder dd has some friends there who think it's great and I know that their mums are really pleased with it so that's also an option.
There are definitely plenty of options for playgroups and things but that is because there has been a real increase (and related large house price rises) of families being priced out of Richmond/Twickenham/Teddington).
On the downside outside of the parks and playgroups there isn't much going on in Hampton it's very, very suburban so if you're used to a lot of life it might seem seem quite quiet? Hampton Hill is admittedly better for that and not sure of your budget but if you wanted a few more shop/restaurant options on your doorstep you consider Teddington?
In terms of roughter/nicer areas Hampton Hill is all nice. Hampton village (around the station) is also all nice. As you move closer to the border with Hanworth either down Swan Road or down past the middle of Hanworth Road and behind the senior schools these aren't so great. Nurserylands used to not have a great reputation and the bit around the small Sainsbury's isn't amazing but I think it's getting nicer.
HTH and good luck, I've probably been a little bit pessmiistic as it really is a nice place to raise a family.