To the contrary, as a resident I'd say Florida's very best offerings are nearly all outside of the tourist areas. You just don't know where to go and/or you're not an outdoors type person.

The Florida state parks site is a great suggestion. I'd also suggest getting away from the east coast if you want to avoid overcrowded tourist traps. Visit the quieter, wilder Gulf coast to see more of the "real" Florida. If you have the time, I'd recommend a visit to the Big Bend area between Cedar Key and Appalachicola. Lots of unspoiled paradise.