If you have seen pictures of Puglia online, they are likely of the famous beach cove in Poglinano a Mare. Poglinano a Mare is a walkable town known for its artists, seafood, gelato, and of course stunning beaches. Visit for the day or spend a couple of nights here. If you do spend the night, we recommend the accommodation Dimora Talenti: https://www.dimoratalenti.it/
This place is simply a dream - stayed here a couple of nights in 2019 and cannot stop thinking about it. Central location to visit other Puglian towns. The owners are lovely, and if you do stay here be sure to do the olive oil tasting with them.
Ostuni is a must when visiting Puglia. This beautiful town sits on a hilltop and is only a 20-30 minute drive from Fasano. Recommend staying at a Masseria in Fasano or nearby and visiting for the day or evening. Make sure to get a reservation at Osteria de Tempo Perso - a charming restaurant at the top of the hill, you won't regret it.
Lecce is a historic walled city known as the Florence of the South. It is the second most populated town in Puglia and is known for its Baroque architecture. Lecce is the perfect place to stroll and explore, with great people-watching and restaurants. Recommend visiting for the day or staying overnight.