Looking for how to get from the Tampa airport to Sarasota. (Without renting a car)
What are the options?
Without renting a car, or taking a very-expensive taxi for the 70-mile-plus drive, an alternative is the Greyhound bus. The downtown Tampa station is at 610 East Polk Street, and you can make your way there, using the Hillsborough Area Regional Transit bus that leaves from the TIA Terminal.
Another option is the Super Shuttle (a personal preference among airport shuttles): a rough estimate of the cost is under $100; your specific destination would affect that. You can book online, as well as pay in advance, for both the ride and gratuity. While more than the bus, it takes you door-to-door, expeditiously, and without any other costs for ground transfers.
Rome2Rio.com is a useful tool for this kind of search, as this was, and you can refer to it for detailed bus numbers, routes, and times.
Despite my initial skepticism on this matter (see the comments above), it does turn out to be possible to take public transit from the Tampa airport to Sarasota, via six (!) separate buses:
Hillsborough Area Rapid Transit (HART), Route 30, from TPA to Marion Transit Center in downtown Tampa.
Pinellas Suncoast Transit Authority (PSTA), Route 100x, from Marion Transit Center to Gateway Mall in St. Petersburg.
A combination of PSTA buses (at least two) from Gateway Mall to Grand Central Station in St. Petersburg.
Manatee County Transit, Route 203x ("Skyway Connexion") from Grand Central Station to DeSoto Station in Brandenton.
Manatee County Transit, Route 99 from DeSoto Station to downtown Sarasota.
Note that the Skyway Connexion bus only runs twice a day on weekdays (10:58 AM & 3:58 PM from Grand Central), and does not run at all on weekends. If those times doesn't work with your incoming flight time, you're probably out of luck.
This whole thing would be a lot easier if there was a direct connection between the Manatee County (Bradenton) & Hillsborough County (Tampa) bus systems; but as far as I can tell, no such connection exists.