I forgot Matera (Basilicata region, but near Puglia). That can also be visited in a day trip, although many prefer to stay at least one night.
Be aware that your car cannot enter the Sassi area of Matera (the highlight of the town), so you need to park outside and if you stay overnight you also need to find a hotel outside the Sassi (or leave the car outside in a garage).
Matera is about 1h10 min from Alberobello and about 1h20 min from Monopoli.
I would choose as few bases as possible, because your vacation is very short (10 days, maybe even fewer nights) and changing hotels is time consuming.
If you choose accommodations in the central Puglia area (Alberobello/Locorotondo, or on the coast Polignano/Monopoli) you can see it all from one place.
The only exceptions are Lecce/Otranto/Gallipoli, and also Brindisi (if you insist on seeing Brindisi). These are kind of too far south, so it should have its own base.
If you plan on visiting Vieste (Gargano peninsula), then I suggest a separate base for that. Gargano roads are very curvy and it takes forever to get there, so you can't easily visit it on a day trip.
Tropea, suggested above, is not in Puglia, it's in Calabria, and it is way too far to even be considered for your trip. It's like 5 to 6 hours to drive there.