Puglia is a long, narrow region with lots of coastline and you'll find good beaches with clean water pretty much everywhere in Puglia. If you don't have a lot of time, it's best to concentrate on just one part of the region as there's a lot to see everywhere.
In southern Puglia, visit the Salento Peninsula, including the Baroque city of Lecce and the seaside towns of Gallipoli and Otranto, and Alberobello where there's a large concentration of trulli, the unique conical structures famous in this part of Puglia. Alberobello is probably the most touristy spot in Puglia but also one of the most unique.
If you don't want to rent a car, Puglia does have public transportation options to many places. The main Italian rail line runs inland to Foggia, then along the coast all the way to Lecce. From Lecce you can get to some places on the Salento Peninsula on a small, private rail line.