If you plan on visiting Taranto make sure you thoroughly research your train/bus schedule there AND BACK! We left from Matera in the early afternoon assuming that we'd be able to stay there until at least around six or seven, and it just so happened that the bus we came in on was the second to last bus leaving Taranto that day. The last bus lefft at around 4pm. We ended up only having about two hours there, most of which was spent looking for the bus (it didn't leave from the same place it dropped us off).
The near airport are in the city of Bari (80 km) and Brindisi (90)
There is the higway A14 that go from Bologna to Taranto (about 720 km)
Surely you need a car to get around Taranto because there are not a great public service of bus and train.