In Puglia there are airports of Bari Palese and Brindisi Papola Casale. You can arrive to the respective central station from the airport of Bari by new metro station and from Brindisi by bus shuttle.
From Taranto take the SS7 Via Appia Taranto-Brindisi, take the exit to Oria and follow the signs to Manduria.
In Manduria there is a Ferrovie del Sud Est station, managed by Trenitalia. By Trenitalia official site you can choose itinerary and costs, as well as choose if you want to use bus or train.