Hitching on the interstate is possible, but dont expect long rides. Florida police wont bother you if you stay on the on ramps of the interstate (turnpike no idea). in the county of the florida keys (monroe?) hitchhiking is illegal though and you may have a hard time. there is a bus though from homestead to key west for only 3 bucks.
Hitchhikers are rare in Florida, but it's still possible to get rides. It's best to stick to the Interstates; you won't have much luck on US highways and State Roads because so much of Florida is suburban, and most people on these roads aren't going far.
If you want something fun to do outdoors, going to a spring can be fun. They're all over the state, very cheap, and usually not very crowded. They can sometimes be off the beaten path, though.