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Florida has slim pickings as far as /out/ is concerned, thanks to massive overdevelopment. t. former Tampa guy
I'd start with state parks. chances are you're going to have to reserve spots a week or two in advance. The only real exception to this for me was Ocala National Forest where dispersed camping is allowed. It's very far from you but maybe there is something similar near you.
But generally reserving a campsite for primitive camping is a good way to get comfortable in the woods starting out. I recommend Myakka State Park if you're on the west coast. And ofc if you're anywhere in the woods in FL bring DEET (for horseflies and gnats) and sunscreen.