I'd highly recommend going to the Kennedy Space Center. You can still see lots of 1960's technology, rather than the Disney-esque artificial experience at Space Center Houston. Otherwise, my own temptation would be to stay away from the big tourist destinations. To me, the best parts of the east coast of Florida are the amazing beaches and water (try going diving for a day...), and some of the fantastic kitsch. South Beach Miami is practically the art deco capital of the world. Also, your adventure isn't complete without eating lots of weird food. Make sure you find some good Cuban and/or Haitan restaurants.

It's been a long time since I was last in Key West, but I remember it being relatively quiet, and having some very good restaurants. Needless to say, eat the Key Lime Pie early and often.

I've never been to Tampa or anywhere else on Florida's gulf coast. I'm told that, while the east coast is populated with retired northeasteners, the west coast is populated with retired midwesterners. I suppose that means you'll find less good deli in Tampa, but people will be noticably more polite. Of course, the Spanish-speaking population is sufficiently strong in Florida that they really set the tone for many things.

If you're looking for other things to do, and you're into salsa dancing, you should know that Miami is one of the biggest cities for salsa. You've got plenty of options there to keep you busy.