The old city (Città Vecchia) is small enough to walk entirely in half a day. The Ponte Girevole connects it to the modern city. Taranto has a local bus network covering the wider urban area, but the main visitor sights in the historic centre and the MArTA museum are all walkable from each other.
Be aware that the Ponte Girevole sometimes closes to pedestrians when it opens to let ships through — the wait is usually short but can be 15–20 minutes. For the beaches south of the city and the Parco Archeologico di Saturo, you'll need a car or taxi. Taxis are available but not abundant in the old town.