We like (in no particular order):
Hanna Park in Jacksonville. It's a city park that feels like a state park. It is within walking/biking distance to the beach. There are hiking and some good mountain biking trails. No pool, but there is a playground.
North Beach Camp Resort in St. Augustine. It's a private park with a state park feel. The sites we've been in are fully private. There is a small pool and it is across the street from the beach. It's a short drive to downtown.
Fort Wilderness in Disney World. It has everything you asked for and more. It's really pricey, but definitely worth a visit. Our favorite campground. We rarely visit the parks, just spend our days at the Fort.
Tropical Palms in Kissimmee. It's about as close to Disney as you can get. It's a really nice park with lots of planned activities. The pools are very nice and the little one will love the kiddie pool play area. We've had deer walk right up to our site in the grass spots.
Henderson Beach SP in Destin. One of our favorite state parks. It's in the center of Destin. It's a short walk to the beach. Sites are pretty private and are roomy.
Topsail Hill SP in Santa Rosa Beach as far as state parks go it's a dud, but still a nice campground. This is like a parking lot campground where the line painter was drunk. There are some trees. It's a nice park with a pool, not far from the beach.
-Michael
