From the airport, "Papola Casale" in Brindisi (Click here for times) and train / bus / car hire to Gallipoli.
Motorway A14, Bari Nord exit, continue on the road Bari - Brindisi - Lecce - Gallipoli.
From Lecce station, continue by the local train "Ferrovie Sud-Est" (click here for times) to Gallipoli.