The Kennedy Space Center is a major historical site with extensive visitor facilities, but the visitor center has to pay for its own operations--not supported by NASA or the National Park Service, so it's not quite what you'd expect as a showcase for the nation's accomplishments.
The Space Center is bordered by Merritt Island National Wildlife Refuge and Canaveral National Seashore. There are fine opportunities to see manatees and birds, but of course you need a car.
Florida doesn't have a long list of pedestrian-friendly beach areas with restaurants and other activities near hotels. Clearwater would qualify (it's part of a large, very spread-out urban area, with some 4 million people), as would parts of Naples, a number of the southeast Florida beaches including Lauderdale by the Sea, (to some extent) St. Augustine's beach, which is really fairly quiet but very good for wandering around, New Smyrna Beach (if you're close to Flagler Avenue, the high street), and my own town, whose pedestrian-oriented beach shopping area tends to service wealthy retirees from places like New York City.
For wildlife, alligators are mostly best seen in captivity. If you take the boat tour at Blue Spring State Park north of Orlando, which goes along the St. Johns River (or boat tours in other parks), you're very likely to see alligators. It's difficult to see crocodiles in the wild. They're rare and shy. You can find advice on where to see manatees in most coastal areas. Dolphins are fairly common in many areas, though there's been a lot of deaths lately, possibly from polluted water in Florida and from a virus disease in the northeastern states.
Florida has spectacular birds. With just a little planning, you can see sandhill cranes, bald eagles, lots and lots of ospreys, pelicans (white ones in winter), caracaras, ibises, and roseate spoonbills. There's a number of Environmental Learning Centers around the state, and some of them provide great opportunities to see the local critters.
The northern part of the Florida peninsula has its own nature attractions, but the main cities (Daytona Beach and Jacksonville) are not exactly tourist destinations, and I don't think Gainesville, home of the huge University of Florida, is much thought of as a tourist destination, either.
Looking at the US as a whole, I'm sure destinations will be constrained a bit by convenient airports, air fares, and the interests of the kids and Mom. California has a bunch of nice coastal towns, including the rather big city of San Diego, which has one of the best zoos anywhere. A movie from last year, "Chasing Mavericks" gives a reasonably good portrait of Santa Cruz.
In terms of cities near beaches on the Atlantic coast, the western portion of Florida is weak on cities, strong on rampant development of coastal hotels and vacation homes. The region has an impressively rich flora and fauna, including spectacular pitcher plants. For more insect-eaters, Wilmington, North Carolina has become a fairly nice city over the past 30 or 40 years (the region was impoverished back around 1960, as were the South Carolina and Georgia coasts). Charleston was a grand city from the 1700s through the 1860s, with enough left to be a very busy tourist attraction. It has nice beaches nearby but murky water and little surf. Roughly the same with Savannah, which was a busy cottom port.