We just spent 3 weeks in Puglia in November and December (LOTS of rain in Lecce a month ago), and, besides Lecce, stayed in Altamura and Martina Franca. Both were excellent towns themselves, and as vases for exploring further. Seeing all the decorations leading up to Christmas was a bonus.
We had a rental car, and I’m glad that you’ll have one, for ease of reaching places. We didn’t visit the beach, or explore south of Lecce. Do be aware of ZTL zones in the center of many of the cities and towns, which have limitations or exclusions for cars, subject to hefty tickets and fines.
Altamura, “The Bread City,” does have outstanding bread in the bakeries, and visiting the Antica Forno, with the giant 700-year-old wood burning oven still in use, is a must-see for its history and the bread and goodies its bakery produces.
Martina Franca’s old town is a charming and atmospheric maze of narrow passageways, leading (eventually) to a grand square. There’s a more modern town outside the old town with its walls and towers. We were able to use a free parking lot - limited number of parking spaces, but in early December, we always got one - and ask your lodging about where to park, hopefully convenient and not expensive.
You’ll likely see trulli (cone-roofed houses and buildings) many places, but Alberobello is the town most associated with trulli. It’s gotten quite touristy, but a must-see in this UNESCO town is the Trullo Sovrano museum, very historic, and the only 2-story trullo. Lots of modern buildings appear to be getting trulli cones put on top of them, but this one genuine has to be seen.
Other towns that were charming, and worth visiting, but I won’t insist that any are “must-sees” for everybody, include Locorotondo, Cisternino, Ostuni, Nardó, and Gallipoli.