Bari Karol Wojtyła Airport: Use the direct Pugliairbus-style airport coach connection to Taranto when its timetable fits; it is the sensible no-car option, roughly 1½ hours and about €10–15. Bari has more flights than Brindisi, but do not assume a train from the airport is quicker: it normally means going via Bari Centrale and adding a change.
Brindisi Salento Airport: Brindisi is geographically closer. Take the scheduled airport shuttle/coach to Taranto rather than paying for a taxi; allow roughly 1¼–1½ hours and around €10–15, subject to the limited timetable. It is the better arrival airport only if the coach timing works.
Taranto railway station: The station is on the mainland, a short walk north of the old-town island; use it if arriving on a through Trenitalia service from Bari, Rome, Bologna or Milan. For most central hotels and MARTA, walk or take an AMAT bus rather than looking for a taxi.
Taranto bus terminal, Via Cristoforo Colombo: Long-distance coaches stop by the commercial port and near the bridges into the centre. It is a practical arrival point for FlixBus, Marino and other intercity operators, within walking distance of the Borgo Umbertino and Taranto Vecchia.