I've posted versions of this list here before, but tried to tailor this one to traveling with kids (we took our grandchildren to Puglia one spring).
The search on this forum isn't the best --- instead try Googling on "Rick Steves Travel Forum Puglia" to see lots of recent posts.
Alberobello — Do see both the touristy side (just try to imagine it without all the shops!) and the other side. Rione Monti is crammed with shops selling trulli knick-knacks displayed out in the streets every few feet and really spoiling the look of the place. Walk through and try to imagine it all gone. Go to the Aia Piccola rione for a less commercialized experience and to the terraced public gardens across the main street for the best view of the Rione Monti. Eat somewhere else, like in nearby Noci. You will see trulli everywhere in this area while driving around, ones both un-used and fixed up.
Bari --- Quartiere delle Orecchiette. Also just explore the maze of the old town generally. Cathedrals, of course.
Ceglie Messapica — piazza was fun in the evening, several good places to eat.
Cisternino — cute little town.
Lecce — we stayed 5 nights in the old town but near a large parking lot, has maybe my favorite church exterior (and kids like the animals, mermaids, dragons, eagles, etc), many places to eat, Roman ruins in the central piazza, great gelato, a fun and lively town to stay in (maybe the best?), visit the transportation museum!
Matera in Basilicata —- Try to walk all over the sassi area —- there is nowhere like it. Sasso Barisano (being renovated, some shops and restaurants) and Sasso Caveoso (wilder, more like it used to be). We enjoyed a little museum full of stuff collected from the old cave-dwelling way of life. Great for kids.
Nardò — small town with great bread and focaccia. Piazza Mercato, Piazza Antonio Salandra
Ostuni — we stayed 6 nights in the old town, great places to eat, LOVE Ostuni. No cars in the oldest part of town, so nice for kids. Our apartment held the 6 of us nicely and I can look it up for you. Can be tricky to find a place to park.
Otranto —— maybe nicer and more interesting than Gallipoli if you have to choose one . Mosaic floor of its cathedral (wild & crazy people and animals and mythical creatures).
Polignano a Mare — a couple hours was enough for us, tiny medieval center on the edge of the limestone cliffs. Small pebble beach. (Crammed with people in summer and seemed rather tourist-focussed even in March, compared to other towns!).
Trani — we stayed here for 4 nights, harbor, cathedral, places to eat, fun passeggiata by the sea, with fishermen unloading their wares. Really interesting old town, including medieval Jewish section.
Two B&Bs we loved (we now rent apartments in towns or a trullo in the countryside)
Locorotondo ----- Masseria Aprile (one of the nicest places we’ve stayed at in Italy)
Matera ---- La Corte dei Pastori B&B (in Basilicata but so easy to visit from Puglia)
Pugliese restaurants we especially liked
Osteria Monacelle (favorite place on this trip, very very small) in Ostuni
Vicolo 45 in Ostuni (it's in a cave)
Osteria del Tempo Perso in Ostuni
Trattoria Fiori di Zucca in Lecce (outside the walls)
Pizzeria Doppio Zero in Cisternino (different ancient grains used in the dough)
L’Antica Locanda in Noci
Towns in Puglia do tend to close up for most of the afternoon, maybe noon to 4:00pm, even the churches. We like it that people prioritize family and eating meals at home over being convenient to tourists. We spent this time exploring the old town centers before and after lunch, and eating long restaurant lunches from 1:00pm to 3:00pm. Always order the house antipasti!
