Honeymoon Island is one of the jewels of Florida's state parks system. You take the Dunedin Causeway over to the island, most of which is the state park. The entrance fee is $5 per car - well worth it.
The beaches are delightful, with plenty of parking, although the middle of the beach is often crowded. Be smart and head off toward the north end, where you can walk a ways and practically have a whole stretch of beach to yourself. The small concession area has a fantastic "breakfast on the beach" on Saturday mornings, with reasonably priced, great food, and you can't beat the view!
Honeymoon Island also has great trails--just follow the main road north and you'll run into them. You probably want to visit the trails during fall, winter, or spring, because during the summer, the trails can be oppressively hot. You'll see lots of wildlife: last time I took the trail, I saw a very large snake.
Probably my favorite part of Honeymoon Island is the FANTASTIC dog beach. The south end of the island is reserved for dogs, and contrary to what you might think, it's not an ugly, trashy part of the beach. In fact, I think it's the prettiest part of the island, with lots of white sand and great views. I believe they reserved this part for the dogs because it is closest to the boating channel: since dogs have to be on leashes, it's not as though they will swim out the 100 yards or so into the channel. Lots of non-dog owners frequent this beach to watch the dogs frolicking in the water. They've even got a doggy shower station so you can rinse off your pup when you're done.