There are several large malls in the Tampa Bay area, and the best for browsing a huge variety of shops is the International Plaza close to the airport. The Bay Walk Mall is newer, and has plenty of chain stores inside it as well as a large theatre. The Bay Walk Mall is open air, so be sure to wear sun cream when you're going round it.
There are several flea markets that appear around the city at weekends, and one of the best known is Wagonwheel, which has a good reputation for bargains on antiques and handicrafts.