A fantastic organization that is part aquarium and part rehabilitation center/laboratory. They just broke ground on a new building, so check their website to make sure if the exhibits you want to see are available.
A cute little park on the water near downtown. Inside the park, O'Leary's is a great tiki bar to grab some cheap eats, and nearby, Marina Jack's is another choice if you're feeling slightly fancier. The famous sculpture, "Unconditional Surrender" is nearby as well.
Kayak through the mangroves with or without a guide. Getting on the water is the best way to experience Sarasota, and with a kayak or paddleboard, it's super-easy and affordable. This is just one of many companies that offer this service.
If you want to see Alligators en masse, this is the place. Observe from a safe distance or get in a canoe and paddle around in gator-infested waters. I don't know why I ever did this as a kid, it was terrifying. But it's an option?
Farther away than most activities, The Dali in St. Petersburg is worth it if you have the time. It's such a cool museum. It has many originals, with one standout being the GIGantic 'Hallucinogenic Toreador' in their collection.