Tampa is primarily a car city, and without a car you will find it difficult to reach Busch Gardens, the beaches, or most of the suburbs. That said, the historic core — downtown, the Riverwalk, and Ybor City — is very walkable and easily covered on foot or by bicycle.
The TECO Line Streetcar runs between downtown and Ybor City along a heritage route — a fun, slow way to make the journey between the two districts. It runs roughly every 20 minutes and is free for most of its route. Walking the same distance takes about 25 minutes.
The Riverwalk connects the major downtown attractions (Tampa Bay History Center, Florida Aquarium, Curtis Hixon Park) along the waterfront and is the best way to move between them.
Rideshare (Uber and Lyft) is widely available and inexpensive by most city standards. Taxis also operate but are less ubiquitous. For getting to the beaches (St. Pete Beach, Clearwater, Treasure Island), car hire is the practical option — the public bus connection is technically possible but very slow.
Bikes are available for hire through the Coast Bike Share scheme, with stations at major downtown locations and the Riverwalk.