Weeki Watchee for the mermaid show is a can't miss.
Gatorland is a must see.
In Central FL (more old FL than wild and weird): Mt Dora is a cool artists community, paddle boats on Lake Eola in Orlando, tons of great Cuban food, Winter Park scenic boat tours at that place on Morse Ave, see Cypress trees/knees at Azalea Gardens or Mead Gardens in Winter Park, Beefy King in Orlando or Kappy's in Maitland for an old school dive sandwich place. Not sure if T.G.Lee Dairy still does tours. The Winter Park Library is one of the coolest examples of MCM architecture. The Milk District in Orlando is funky with some off the beaten path shops and restaurants.
St. Augustine tours - very interesting history and architecture. Some pirate themed stuff.
Key West for some weird stuff on Duval St at sunset.
Daytona Speedway for watching races. It was very weird to me at least
Kennedy Space Center is great.
Ron Jon Surf Shop in Cocoa Beach.
Have fun!!!