By far, the best way to get to and from the airport is the Super Shuttle. It picks up and deposits fares from both Tampa International (TIA) and St. Petersburg/Clearwater International (PIE).

If you can, renting a car at the airport is your best bet for arrival, departure and getting around in general.  There are a lot of things to do in the St. Pete/Tampa area and transportation is not easy unless you want to limit yourself to a specific area -- if you do -- choose one, the beach or downtown.  The in betweens are very suburban.