Bari Karol Wojtyła Airport: Take a direct Cotrap/Miccolis-style airport coach to Taranto rather than detouring through Bari Centrale. Expect about 1 hr 25 min to 2 hr depending on the run and roughly €10–20; schedules are sparse, so match your flight to a departure before committing. Bari has far more flights than Brindisi and is usually the practical choice.
Brindisi Airport (Papola Casale): Use the direct Pugliairbus/airport-coach connection for Taranto when its timetable works: roughly 1 hr 15–1 hr 30 and about €10–15. It is closer than Bari, but has fewer flights and fewer useful departures. Do not take an airport taxi for this intercity trip.
Taranto railway station: The station is on the mainland, a short walk north of the old-town island. Direct long-distance trains connect Taranto with Bari, Bologna, Milan and Rome; use it if your itinerary already starts in Italy, but check whether a faster connection via Bari beats a nominally direct service. A taxi or local Kyma Mobilità/AMAT bus is sensible with luggage; central sights are otherwise walkable.