Getting to Bari/Puglia

I flew into Bari. In addition to Trenitalia, there is a local railway that also operates buses: https://www.fseonline.it/
I'd start there. If you have ample time (and patience), a public transport exploration of the area is possible.
Some of the airport transfer may also be helpful: http://www.aeroportidipuglia.it/en/web/10181/bus-pugliairbus-
(there is an airport bus to Matera at certain times)
I find the website RometoRio helpful for finding out how to get place to place.
good luck!

In addition to Southeast Railways above (Bari–Alberobello–Taranto, tickets about $10), a couple of other local rail operators are Gargano Railways (www.ferroviedelgargano.com/en, from San Severo) and Bari–Matera trains by Ferrovie Appulo Lucane (ferrovieappulolucane.it/en, about $5).

I visited Puglia in 2015. I used Lecce (a very pretty baroque city) as my base, which proved beyond fortuitous, because the tourist office in Lecce arranges van trips to small towns in the area. They basically just provide transportation to multiple towns each day, typically 4 to 6, so there wasn't a lot of time in each place. But the price was right. The driver would get us near the historic district, point out a major sight or two, and tell us when to be back at the vehicle. These are not fully-guided trips; given how fast they move on to the next little town, that works better, since everyone can walk around at his own pace. Take maps (paper or electronic) with you.

The trips were a great opportunity to see a lot of small places in a relatively short time. One included Matera; all the other destinations were in Puglia. They ranged as far north as Polignano al Mare.

The trips depended on demand, so I didn't usually know more than 24 hours in advance where I'd be going the next day, and there isn't necessarily a trip every day. Still, this was far easier than struggling to get to even two places a day via public transportation.

I can't find an English version of the website, but these are the trips currently listed. I see the price is 60 euros for the full-day trips. I didn't take any of the half-day trips (40 euros).

T.O. staffers spoke English well. You could try emailing them at [email protected], but I'm not sure they'll be in a position to do more than confirm that they plan to continue the van trips this summer.

Matera, in particular, would benefit from more time than you'd get on one of those trips, so you might make it a stop on the way to Puglia and spend a night or two there.

For train schedules, check TrenItalia in addition to the other rail companies already mentioned. Rail service is fairly limited, so beyond the van trips you'd need buses to reach many of the picturesque small towns. Rome2Rio is a good source for determining whether train or bus service is available between any two towns and which company offers it. I find the fare and trip-duration information not terribly accurate, but if you keep drilling down for bus links, you'll usually come up on a link to the appropriate bus company's website. If not, try Googling "Bus Lecce to Alberobello", etc. That will often turn up a schedule.

Bus service tends to shrink or disappear entirely on Sundays and holidays, so be very careful about planning your itinerary on those days.

You might consider also staying in a second city in Puglia that's farther north. I've heard Bari is nice. I don't know whether Bari has an equivalent of Lecce's van trips, though.

Incidentally, the summer heat in Puglia can be brutal. Prepare yourself mentally.

We did several weeks in the far south, all by public transportation. It's not necessarily quick, but we were able to get (almost) everywhere we wanted to go in our allotted time. The best source I found for transportation information was in the back of the the Lonely Planet guide to Puglia/Basilicata. (We went before Rome2Rio became a "thing"; I certainly think that site is very helpful also.)

Thank you all so very much, your info and suggestions will be a great help!