Puggia

I tried to insert Vieste into two (albeit shorter) trips while limiting bases, and I just did not find it possible--it is hours away from anything else on your list. But you do have time to include it, it would just have to be a three-base trip.
For the rest: One base in the Itria Valley (*or adjacent coast) and one in the Salento, though I also argue that Matera should get its own stay as well.

The region has a long shape, and public transportation is not too fast. The two main railway lines (Foggia-Bari-Brindisi-Lecce and Bari-Taranto) are workable, but the minor lines (a dense network linking the interior and Matera) are very slow. Having a car could be a plus but your plan of having two bases and a lot of places to visit could translate in a lot of driving times. - I second the idea that Vieste and the Gargano area is too distant for day trips from other places.

It's spelled PUGLIA and is pronounced PU-llia (like Spanish LL as in "LLAVE", similar to MILLION in English)

For the Gargano Peninsula, Vieste or Peschici are good bases. However getting out of there is time consuming due to the nature of the roads, and as such it can be used as a base to visit the Gargano only.

Alberobello and nearby Locorotondo, are in the same area and central enough to be used as a base to visit also Matera, Ostuni, Martina Franca and Polignano. You can also use Polignano as a base. Not as central, but if you prefer to stay at the beach, you can use it.

Lecce, Gallipoli, and Otranto are kind of farther south, so it would be at least a couple of hours to reach from Alberobello or Polignano. Maybe you should add another base in the Lecce province (Lecce, Otranto, Gallipoli are all good.

Sorry for the mispelling, English is not my native language. With your commentaries, we are now thinking to skip the Vieste area and concentrate around Bari and Lecce. Day trips of one hour or lest from these two base