I live in Tampa.
St. Pete is having a hipster renaissance of sorts. The downtown is becoming very popular for the 25-35 year old crowd. Tons of new clubs/restaurants/breweries opening there and the night life in general has really picked up.
St. Pete also has some really bad areas. It used to be the homeless area of choice but they cleaned that all up a few years ago and basically they all moved over to Tampa.
Schools? No clue. But I imagine like anywhere - find an area with a good income bracket and schools will generally follow. Clearwater I don't think is really known for any good schools. When I think of Clearwater I think beach town.
It really comes down to how much you have to spend. Tampa you can find a great 3 bedroom house for 200k-350k in a nice gated suburban neighborhood away from any crime really. You will end up having a drive though. If you want to live downtown in a highrise it's gonna be 200k for a 1 bedroom and there is no existent mass public transport except for the laughably bad HART bus system.
For Tampa - look into Hyde Park, Lutz, Carrollwood, South Tampa, Wesley Chapel, New Tampa. Those are the ones off the top of my head with either a population that earns a high income or is middle class / good schools. Some of those will be quite a drive to St. Pete / Clearwater though. Some wont be.
Tampa is not a beautiful city by any means. I call it the New Jersey of Florida. Miami the New York and Orlando the LA.