There are 2 main airports in the Tampa Bay area.
Tampa International Airport
Tampa International Airport (TPA) is the main airport for Tampa Bay and is located on the west side of Tampa, about 30-40 minutes from Dunedin. Tampa International is one of the best big-city airports I have been too and is much easier and faster to get around than airports like Chicago, Atlanta or Los Angeles.
The main or “Landside” terminal where you find ticketing, baggage claims, dropoffs and pickups is also right next to the long-term parking garage and there is also a Marriott hotel attached to it that you can walk to right past the shops and restaurants in the main terminal.
The gates are in separate “Airside” terminals that surround the main terminal and you get to them by taking a tram. The amount of walking you have to do is significantly less than in most other airports.
St. Petersburg-Clearwater International Airport
The other airport is the St. Petersburg-Clearwater International Airport (PIE) it is a much smaller airport than Tampa International and only has flights to a handful of Midwest and East Coast cities. The 2 main airlines that fly out of this airport are Allegiant and Frontier although there are also some smaller airlines that have flights here too.
If the city you are flying to/from has a direct flight into or out of this airport you may want to consider using it because often the flights are less expensive and it can be easier getting in and out since this is a smaller airport and handles less flights and passengers.