3 days/3 nights in Puglia

I would stay in Matera, especially if you have a car. There are numerous cave hotels, a couple of museums, the incomparable Sassi, and the nightly passeggiata along the main street. Some great restaurants, too. You are within easy distance of Puglia, even though Matera is in Basilicata.

You won't need a car in Matera, and parking can be difficult. The streets are very narrow, winding, and maze-like. So rather than a train to Bari and renting a car to drive to Matera, I would take a bus directly from Rome to Matera. Use Rome2Rio to check all transportation options.

Stay in Matera one night, in a cave hotel, and the next day rent a car on the outskirts of Matera and spend your remaining two nights either in Alberobello itself, or in a masseria (there are lots of them around Fasano). Or you could even stay in a pretty little town along the coast (like Polignano a Mar) and still be only 30 minutes away from Alberobello.

After your three nights, it's a short trip to Bari to return the car and you can take a train back to Rome.

Parking in Matera: We found free parking on the street near the castle, and in a small lot in the Sassi (spaces there go fast, but we never had a problem finding a space when we returned from a daytrip in time for dinner). There are also several pay-to-park facilities in the area.

You cannot drive down some streets in the Sassi, but it's a better walk, anyway. When in Matera itself, we walked everywhere.

I'd prefer to take the train from Rome to Puglia. While Matera sounds/looks interesting - my focus was going to be Puglia. I'm wondering best place to stay for at least for the first 2 nights so we can see the major sights. Will want to pick up a car rental - from the rail station.

We can always drive to Matera for the last night - our flight is from Bari at 940am to Paris.
Looking for advice on how best to spend the 2.5 days/3 nights there without getting crazy - don't' need to see everything.

I rented a Truli in Cisternino. Great location near all the great Pulian towns. However you should rent a car. I took a train from Cisternino to Lecce. Lecce is a long day trip by car. Take a train and enjoy the countryside.

Lorie

The towns along the coast north and south of Bari would work (i.e., Trani, Polignano, Monopoli).
Also, any of the trulli/Itria Valley towns would work.
Either of these would put you near the airport/train station as well as centrally located for your day trips.

Ok so here is what I really need:
26May - we take the a.m. train to, not sure yet, but probably arrive 130pmish.
We will have 3 nights 2.5 days. Want to be centrally located for touring the Trulli homes, etc. with a car rental. What are the highlights that we can comfortably "squeeze in"? This is a new destination to me - can't get a grasp yet...
29May - we fly to Paris - can be from Bari or Brindisi

My friends and I rented rooms at a small agriturismo about 10 minutes' drive from Bari, called Podere Papilio. The main building is a trullo, with bedrooms for guests attached (separate entrances), each with a small front porch. The owners were planning to expand the property and add another trullo suite. It was a lovely time - fruit orchards, homemade preserves, butterflies, wonderful hosts. Parking inside a gate that gets locked at night (arrange for your arrival if late).

Ok so here is what I really need:
26May - we take the a.m. train to, not sure yet, but probably arrive 130pmish.
We will have 3 nights 2.5 days. Want to be centrally located for touring the Trulli homes, etc. with a car rental. What are the highlights that we can comfortably "squeeze in"? This is a new destination to me - can't get a grasp yet...
29May - we fly to Paris - can be from Bari or Brindisi

Perhaps you should clarify if you want to stay in a town or in a rural place (masseria or trullo rental).
Honestly you could spend weeks in the area--there are lots of tiny towns with one impressive church here, one archaeological site there, etc. It just depends on your interests. It sounds like the trulli is your priority, so any of the towns will do--some are so close together you can walk from one to another. The area is roughly an hor from Matera and an hour from Lecce. The thing about Lecce is you will want to arrive quite early (or in the evening) as all towns shut down roughly from 12-5.
I wold make Matera a priority just because 1) I fear mass tourism will change it soon and 2) it is in Basilicata so a bit of an outlier--flights to Bari or Brindisi for the southern tip may come along again but for now you are going to be so close, and 3) It is really quite impressive.

I've been advised to stay around the Fasano area - can visit Monopoli, Alberobella, Matera, etc. from a Masseria here.
If an overnight is recommended in Matera - any hotel recommends?

To get a grasp of this area, I suggest you look at Rome2Rio, plug in various "to" and "from" cities, and then look at the map that is displayed. You'll see distances, and the many choices you have within an hour of Bari. So it comes down to your personal preferences ... a seaside town, a masseria? One base with daytrips? Or one night in Matera and then two nights somewhere else?

You can easily day-trip into Matera, but I would recommend spending a night. It's such an unusual, special place.

But if you're spending a night in Matera, you might want to reconsider the bus. It would take you directly from Rome to Matera, as compared to taking the train to Bari, renting a car, driving to Matera. You won't need or want a car in Matera. And you can easily pick up a rental car on the outskirts of town as you leave. I've never taken one of these buses but they look nice.

In Matera, we stayed at Locanda di San Martino. Beautiful rooms, and a gorgeous underground thermal spa. It's located inside the historic Sassi area, where there is no parking. But there is public parking available outside the Sassi and a shuttle to the hotel.

Near Fasano, we stayed at Masseria Montenapoleone. Great central location for visiting the trulli villages and the coast. Very quiet and beautiful but you have to drive to restaurants (although they do serve dinner there occasionally, or you can even cook in their well-equipped guest kitchen, stocked with herbs, olive oil and food grown on the masseria -- depending on what time of year you're there). They have great breakfasts and a lovely pool. We had some amazing meals in tiny, picturesque Torre Canne, along the coast, 5 to 10 minutes from the masseria.

If you'd rather stay in a seaside town, I'd recommend Polignano a Mare. We didn't stay overnight there but it was beautiful and had restaurants you could walk to.

I would not recommend a day trip to Lecce unless you are early risers and huge fans of baroque architecture. As has been mentioned, pretty much everything in town closes down between 12 or 1 pm, and doesn't reopen until 5 or 6. Lecce is known for its baroque churches (there are well over 100) and you can enter some of them in the afternoon, but it's so deserted. Almost like a ghost town. The only places open after about 1 pm are coffee shops, where you can buy coffee and maybe a pastry. Stores and offices are closed. No restaurants (even pizza) reopen until 8 pm.

You can easily fill two days in Alberobello and the other Trulli villages, as well as Ostuni. And there's a nature reserve along the coast south of Torre Canne, if that type of thing interests you.