We spent 10 days in Puglia last February- March. The weather was great—lots of sunshine, neither too hot nor too cold, and only 1 partial day of rain (but that was a deluge!). I would expect October to be similar.
We flew into Rome and home from Bologna (having worked our way north via Rimini, Ravenna, and Ferrara after Puglia). The trip from Seattle was longer than what you have from the an east Coast, so we did not arrive ag FCO until 6 pm.
We chose to overnight in Rome (in the city, not at the airport) and take the train to Bari the next day (actually going only as far as Barletta that day). We figured,we would have to get into Rome to catch the train anyway, and preferred a real hotel with walking opportunities nearby to night in an airport hotel. So we took a 50€ flat-rate taxi from the airport to our hotel, near the Villa Borghese but only a 10-minute taxi ride from Roma Termini station. We enjoyed a nice quiet dinner at a restaurant nearby (recommended by the desk clerk as well as the bar an who gave us our “welcome drink”—-both said it was where all the staff went for a reliable meal. The next day we had time for a 2-hour walk in the Villa Borghese parklands before taking a taxi to Termini for our 1:15 train. We allowed time to buy sandwiches at the station for lunch on the train.
The direct Frecce train from Termini to Bari takes only 4 hours, and makes only 4 stops: Caserta, Benevento, Foggia, and Barletta. If you wish to continue partway to Bari after landing at FCO, I would highly recommend Caserta. The train ride is just a wee bit over an hour once you get to Roma Termini station. There is a world-class attraction very close to the train station. (The Reggia di Caserta palace and grounds), with nice hotel options right there too, an easy walk from the train (look at Hotel Royal).
https://visitworldheritage.com/en/eu/caserta-royal-palace-and-park-italy/73cf1988-9d13-4658-99f5-2f23a706bc00
We chose to travel the next day, and for an overnight stay the smaller town of Barletta appealed to us more than the port city of Bari. We found a great little guesthouse, with a spacious room with lots of windows and a delightful hostess, Antonella.
https://www.bbliddo.it/en/
Our dinner at the wine bar named Twine was one of the best we have enjoyed in Italy (and we have spent a lot of time in that country). The ambiance, the food, the friendly welcome—everything was perfect. They do not serve dinner until 9pm but it was worth waiting for.
https://www.facebook.com/twineristorantevineria/
The next morning we toured the stern fortress/castle in town before continuing by train to Lecce.
After our time in Puglia, we made our way north. But if you are considering more time in the very south of Italy, whether Sicily elsewhere, you might like to fly out of Naples.