A week in Puglia

It makes a lot of difference whether you plan to rent a car or not.

depends on your interests first. art, architecture, history, nature, beaches, famous tourist destinations, smaller less touristy places, food, etc. next would be bases where you want to stay and the day trips you may want to make. only then consider how you will get to places and the time to go back and forth. self-drive or public transportation? i stayed in bari and lecce and used public transportation exclusively.

My major concern is the weather, at this point. Don’t want to spend a week in rain.
If we choose a week in Puglia mid November, where should our base be? We would use public transport.

I personally would recommend Lecce, as it has many beautiful sites, great restaurants and decent train connections to other places in the region. Bari would give you the absolute best train connections, though. We didn't spend any time in Bari; loved visiting Taranto and Gallipoli. Had no interest in trulli, which puts us in the minority among Puglia tourists.