Matera and Bari.

We spent 2 nights in Matera last mid-Oct. and stayed at IL Belvedere and it was great. Would stay here again. It is located in Sasso Caveoso close to Via Bruno Buozzi, the main drag through Sasso C., and it is outside the ZTL zone which was nice as we had a car. Were able to park right outside the front door. It is just a few minutes walk to Via Ridola, the main drag up to Piazza San Francesco. Closeby are the churches of San Pietro Caveoso, Santa Lucia alle Malve and Santa Maria d'Idris. We had room #43 which was a huge superior double cave room. They also have some rooms that are not cave rooms. Upon check in they gave us a map of Matera so didn't even both going to the T.I.
IL Belvedere had a nice breakfast with meats, cheeses, yogurt, cereals, pastries and even chocolate drunk cake--Torta Ubrica (con vino aglianico) cake made with local wine! It was delish! The view from the breakfast terrace is amazing! Soul Kitchen restaurant is just down the street and was a great place to eat. We were lucky as they fit us in but I would recommend reservations. The front desk here was very helpful and on the day we left gave us a sheet with directions on how to drive to the belvedere at the Parco della Murgia Materana. We daytripped to Bari from Trani so can't help you there but enjoyed our day in Bari. Have a great trip!
https://www.hotelbelvedere.matera.it/

Although I don't offhand remember the name of the actual B&B that we stayed in, I strongly recommend staying in the old town area of Bari. It is charming and you will get a true sense of Italian daily life there. There are many B&B's available. You will surely have a wonderful time wherever you stay!

Thanks so much for the suggestions and information! Our trip will be at the end of Sept. We will have five nights to divide between Matera and Bari.

We stayed at the Locanda di San Martino in Matera in one of their cave rooms last year and enjoyed it very much. Clean rooms, large bathroom, good breakfast. Here's a link: http://www.locandadisanmartino.it/?lang=en
Matera is an incredible town to visit, unlike anything else you'll see in Italy. We did not stay in Bari, so can't be of any assistance for that city.

Buon Viaggio

I second Locanda di San Martino. A great hotel in a good area.

We just came back from southern Italy including 3 nights in Matera. We stayed in a very cute inexpensive BnB called http://www.alberodieliana.it/index.php/it/

Located in the old town but not in cave. Owner was very helpful. Breakfasts were a treat.

We spent a few hours in Bari walking around between trains (Matera to Lecce). It’s pleasant enough but not sure it warrants 2 nights. Perhaps stay in one of the small towns along the coast - train service is fast, frequent and cheap.

Thanks to each of you for the information. So appreciate the tips!