Hi, Mike. I'm here in Clearwater.
There is no shortage of aviation options within a short drive north, east, and south. It really just depends on what you're looking for. I trained at PIE but have flown to or from all the local airports. Here is a brief rundown. I'm typing quickly so pardon any mistakes:
PIE: Largest in the area. Busiest Coast Guard Air Station in the country. They have the C130s and helos running day and night. Also a base for Allegiant. Also some Blackhawks based on the field. Also tons of GA. At least 3 or 4 schools with rental options last I counted. Lots of GA out of this airport. I trained and rent out of CAMS Flight. I think this is a great combination of "large" airport, towered operations feel, and GA-friendly. You're regularly mixing it up with the Coast Guard and military and GA and private jets and everything else. But it's almost never so busy that you're waiting in line for takeoff.
Clearwater Airpark: Small town GA feel in the heart of Clearwater. Last I checked, the school there had the lowest beater Cessna wet rate in the area. Non-towered operations. Noise abatement procedures because it's surrounded by neighborhoods (folks would love to close it down). No touch and goes allowed.
Albert Whitted: Really a pretty neat little towered airport. Approaches over the beautiful bay waters for most runways. Great restaurant (Hangar). Home of the first commercial passenger flight...ever. Surrounding St. Pete waterfront is quite fun. Last I checked long ago, plenty of rental options here and rates were average.
All of these have Tampa class B overlying.
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Can't go wrong with any of them, I'd say. If you have specific questions about any of them, let me know. As far as things to see from the air:
The local "practice area" is west over the barrier islands. It's beautiful. Flights over the cities are nice; St. Pete and Tampa. Lakeland, home of Sun 'N Fun, is a short hop east with a museum and restaurant on the field. Venice is a short hop south, on the water, with a nice restaurant that'll pick you up for free in a shuttle. Cedar Key is north on the water; old time Key West feel, can rent a golf cart and the guy will pick you up at the airstrip and you can tool around town. Real short runway, be on your short field A game.
It is truly a great place to enjoy GA. Let me know if you have other questions.