Bari Karol Wojtyła Airport: The sensible public-transport route is the Ferrotramviaria airport train to Bari Centrale (about 17 minutes; roughly €5), then a Ferrovie del Sud Est train towards Putignano/Martina Franca/Taranto to Noci (generally 1 hour 10–30 minutes; roughly €4–6). Allow about 2 hours door to door and check the second leg carefully: this is a rural line, not a turn-up-and-go service. For a late arrival, luggage, or plans to visit masserie and nearby towns, rent a car at BRI instead; the drive is about 1 hour 10 minutes.
Bari Centrale: Use Ferrovie del Sud Est (FSE) services for Noci, rather than assuming Trenitalia will take you all the way. The journey is usually around 75–90 minutes. Noci station is a short walk—roughly 10 minutes—from the historic centre, but not from outlying accommodation.
Brindisi Salento Airport: Brindisi is viable only if flight prices make it worthwhile. There is no clean airport-to-Noci rail transfer: public transport means a bus into Brindisi, rail/bus connections via Bari or Taranto, and too much timetable risk. Rent a car; the drive is about 1 hour 20–40 minutes depending on traffic.