K.M.
I second Meredith; I was going to mention most of the things she already said. Devil's Milhopper will be a good way to burn energy off the kids; it's a long stairway down into a very old sinkhole that's now a nice nature-y area. And it's free!
The Itchetucknee River is fun but if you don't want to commit a whole day to tubing, there are other springs in the area too. Blue Springs comes to mind...sorry, that's the only name I can seem to remember, but there are others.
There's a roadside pull-off point on Hwy 441 alongside Paynes Prairie with a dock that's a great spot to see alligators up close in their natural habitat but still out of jaws' reach. You can also see them at Lake Alice on campus, which is near the bat house. Go right at dusk and you can see all the bats fly out of the house. It's pretty neat. I think it's one of the largest bat colonies in existence, too.
I'd probably pass on Kanapaha Botanical Gardens, unless they've gotten better over the last few years; when I lived there, the gardens were always in poor shape.
The next town south (of any size) is Ocala, and there's a two-sided park there called Silver Springs and Wild Waters. One side is a water park, the other is a cross between a small theme park and a zoo. Much calmer atmosphere than Disney World. Just east of Ocala is the Ocala National Forest, which has some great hiking. Do your research before you go, though, because it's huge. Or you can just drive through the Forest on your way to St. Augustine, which is a tourist trap for sure, but still a neat little town with a lot of history. And beach access, if you don't mind the spring breakers.
If you would rather stay away from the spring breakers, then like other people have said, the Gulf Coast is your best bet. Plus the water will be warmer and calmer, which is good for the little ones. If you want waves, though, you'll have to go to the Atlantic side.
About 60-90 minutes southwest of Gainesville is another tourist trap called Cedar Key. It's a small town, mostly known for their great seafood restaurants with great views of the Gulf. There's a small historical society museum with a lot of interesting artifacts and pictures.
Do check out the Florida State Parks website as well as the National Park Service website. There are a lot of parks, wildlife management areas, etc near Gainesville.
The only other thing that comes to mind is the flea market in Waldo, about 30 minutes northeast of Gainesville. Just don't speed in Waldo. It's a known speed trap.
Hope you have fun! I really miss Gainesville sometimes!