Hi J.,
We moved here 3 years ago, so I'm not really an expert, esp since I work in St Pete and don't have as much time as I would like to explore Tampa's mommy resources. We live in (old) Carrollwood, which is not far from the airport. Someone mentioned Al Lopez park, which is a very nice green space south of Carrollwood, near the stadium. They used to have a great playground, but I think they are redoing it right now.
As far as churches, I don't know what denomination you are interested in, but my mom loves St Lawrence Catholic Church, which happens to be across the street from Al Lopez Park. She says they really make an effort to involve the kids, and they also have a private school. I, personally, am a complete heathen, so I don't really have much to add to that!
Good luck with the move!