Some locations positively demand a car, but merely travelling between towns in the area could be done by bus/train.
In terms of driving, Bari airport is a wonderful place to pick up a car, as the roads around it are lovely and quiet, allowing you the chance to get familiar with the controls, signage, etc. before getting onto the autostrada (the latter got busy for the 3-4 junctions around Bari, but was pleasant apart from that).
I'd describe the driving in the region as
- very easy going in-between the towns, some of the easiest driving in Italy
- Going into or through the towns can be stressful, as few have 'bypasses', so you're driving through a historic town, which can be a little higgledy-piggledy, with building walls forming the edge of the road. Such stressful moments don't last long, but having an attentive passenger with a sat nav would help ease much of that stress. Otherwise you're soon through it and back out onto the open and relaxing roads.
Parking can be a challenge, and whilst we got very lucky on our two visits to Martina Franca, I'm told (and can very much believe) it can be difficult to find a parking space. At Locorotondo we found a nice quiet parking area a short walk away, so that was good for us. Maybe worth planning the route into the towns, taking note of the parking areas.
With a car, it's well worth scouting out local agriturismi (farmstays) as in addition to accommodation, many do satisfying meals, often in the form of a banquet-like progression of traditional dishes emerging from the kitchen. They're convivial affairs and the one we stayed at in Speziale was indeed good, though traditional included tripe for one dish, which neither of us can stomach (
Also in range with a car is Matera, plus some Roman excavations between Speziale and Monopoli and I'll also add a good word in for the somewhat humdrum town of Gravina. It has a large street market (I don't recall which days), some good functional shops and a rather bizarre layout, being split in two by a huge ravine, spanned by (IIRC) two bridges. It felt friendly in the manner of places that don't see many tourists.
Finally, Trani is a gem of a place, built up by the Venetians, so there is interesting architecture. It's got a relaxed enough feel and makes a wonderful place to kill time before dropping the car back at the airport, or indeed a place to get fresh air and a bite to eat if needing to kill time on arrival.