Ah, I recognize your handle via some of the TBL vs BOS main board threads.
Welcome to the Tampa Bay area!
It's hard for me to recommend apartment complexes since I'm a homeowner in the northwest Tampa suburb area, but the area known as Town and Country/Westchase is on the west/northwest side of Tampa and would be not too bad a commute to St. Pete even though it's more at the latitude of Clearwater. Westchase is a bit more upscale-ish, but I think both areas have some nice apartment complexes. Even the small town of Oldsmar which lies between Town 'n Country and Clearwater at the top of Old Tampa Bay is an option.
South Tampa mostly lies between Kennedy/I 275 (leading to one of the main bridges crossing to St. Pete) and Gandy Blvd (leading to the furthest south bridge over to St. Pete) would be even closer but the prices and quality of the area varies quite a bit. Palma Ceia, Hyde Park and such are verrrrry pricey areas, but places closer to or south of Gandy Blvd can be very reasonable but again, vary a lot in quality. The further south of Kennedy you get closer to McDill Air Force Base the cheaper places are unless they're right on the water.
Lots of people commute from Tampa to St. Pete and vice versa so the two main bridges across the Bay can get crowded and it can take up to an hour some days.
Hope this gives you some ideas for the Tampa side of the Bay. I know very little about the actual living on the St. Pete side so hopefully another area resident can help with that.