The train from Bari to Lecce takes about two hours and makes many stops. It is part of Trenitalia so it is easy to get online information and buy tickets. As to the train, best to board early to have a choice of seats and to secure a place for luggage.
Better yet, buy a first class ticket if offered. We went first class from Roma to Bari and will in our upcoming leg to Napoli. Well, almost; an hour outside Napoli we have to switch to a leg that is without first class.