We did quite a few non-Disney things. We just got back yesterday.
My two favourite parks ended up being SeaWorld and Busch Gardens. Not nearly as crowded as Disney's Magic Kingdom or Animal Kingdom. Both SW and BG have great playgrounds for kids, good adult rides and free beer. I have a 4 and 7 year old, and both enjoyed the play grounds. I think kids up to 10 would probably like them. Great big 3 story tall play structures, water areas etc. I liked the coasters. My husband liked the free beer. But while not quite as expensive as Disney, these places are hardly cheap alternatives.
We also went to Old Town. OK, we spent thousands of dollars for big name Florida theme parks, and the kids were clammering to go back to Old Town nightly. Go Figure. Its a little amusement park type place with fair-ground type rides and alot of tacky cheap stores with nonetheless interesting merchandise in them. Admission is free. There is an ice cream store and a coffee shop and a couple restaurants. The ride ticket books are $10 for 20 tickets, which will get you 2 big rides, or 3 or 4 kiddie rides. Its not glitzy but its kind of a nice change to the
mega parks - and as I said, the kids loved it.
Disney also has 3 water parks. We tried out Typhoon Lagoon, and loved it. Its kind of laid back, with lots of sand areas and not too crowded. A nice break from the MK and AK days.
We also found we needed a few days to just do nothing - or close to it. The kids got tired of being dragged around day after day after day. If you're staying somewhere with a half decent pool area, you could consider planning a day or two just around the pool.