Fly to Bari or Brindisi to tour Puglia

Where are you starting from? If staying in Rome (rather than arriving at the airport), it would be easier and faster to take the direct train. That takes only 4 hours. Starting from downtown Rome, by the time you add up travel time to the airport, 2 hours security time, boarding delays, and the actual flight time, plus transport from Bari Airport into the city. . . . I think you would already be in Bari.

But if you wish to fly, your choice depends on what you wish to see in Puglia. Brindisi is closer to Lecce and Otranto; Bari is closer to the lovely coastal towns of Monopoli and Polignano al Mare. The Val d”ItLian towns of Alberobello, Martina Franca and Locorotondo are about halfway in between. But all can be visited in a week or so starting at Bari and moving south from there.

I do not agree that you can scratch the surface of Puglia in one week.
Much will depend on the time of year you plan to travel.

If beaches are not on the agenda you will have more time.

As poster, above, mentioned, plan your destinations and then choose the airport.
You might include Matera in the mix; for me it is among the most fascinating cities of Italy, with some of the best food as well. It is over the Puglian border in Basilicata, a much underrated region of Italy.

Another train option, since you are renting a car, is what we did on one of our trips to Puglia --- it's only a 3 hour direct train ride from Roma Termini to the Foggia train station, and then it's very easy to rent a car near the train station in Foggia. We drove from Foggia to stay in Trani, one of our favorite towns in Puglia.

On another Puglia trip, we flew to Rome FCO and then pretty quickly and easily flew straight from there to Brindisi airport and rented a car. Actually a 9-passenger van. From Brindisi, my husband drove us all to Ostuni (no jet lag because we get ourselves onto Italy time before we leave the US and he sleeps on the entire transatlantic flight ). I don't know if jetlag and driving will be an issue for you. Also, you could be unlucky and your flight to Rome could be late and you'd miss the plane to Bari or Brindisi, so an early morning arrival into FCO is a good idea.

Bari is somewhat closer to Matera in Basilicata, if you are including that wonderful place in your Puglia plans, and there is a bus to Matera that you can take from Bari.

Joyce, Lola mentioned the general locations near those two airports. If you do decide to take the train to that area, instead, I took a direct train from Roma Termini to Caserta (1 hr) and stayed two nights there to enjoy the gorgeous Caserta Reggia & massive garden. Then I took the direct train to Bari to begin my 10 days in the Puglia region. Puglia is a wonderful area! I am hoping to go back.

It would be helpful to have much time they have for Puglia. They said “highlights” so I assumed a short time (like a week). But more time would be much better, especially if they are adding Matera, which needs a minimum of 2-3 nights.

Others are saying a week--is that what you have? Some might say the towns near Bari are the highlights, and other might say the Salento, so it really depends on what you wish to see. However, they are not far apart, so I would base it on price and schedule as well as the places on my list.
What month?