How to see Puglia

It is easier to see Puglia with a car, but it is certainly not impossible to do so by bus and train, and that is what I would suggest.
I would stay a night or two in Matera (there is a direct bus from Bari airport) and the rest in Polignano or Alberobello (or any of the Itria Valley towns). I would save Lecce for a future trip, as it is much further away and may requite a long time on a bus or transfers. You could stay in Bari the last night if your flight is early, but it is not far on the train from Polignano, so you could easily stay there before an afternoon flight.

If you were to fly into Brindisi, you can easily get to Lecce and stay 4-5 days there, taking day trips to Gallipoli and Otranto.
You are likely familiar with Trenitalia, which does offer some service in Puglia, but you need to also familiarize yourself with the other local train lines:
https://www.fseonline.it/s/
https://ferrovieappulolucane.it/
Bari airport bus service: http://www.aeroportidipuglia.it/bus-pugliairbus-

I camped out in Lecce (a beautiful Baroque city) for 5 days or so back in 2015, taking advantage of the van trips (not really tours, just transportation to a bunch of smaller towns) organized by the Lecce tourist office. You had to be flexible, because trips ran only when there was enough demand. I suspect there may not be anything happening right now. In fact, I cannot find the tourist office's website at the moment. That may not be terribly meaningful; that has happened from time to time in the past.

I did find this stripped-down website for a Lecce-based travel agency that advertises tours. If you're studying in Italy, I assume you have enough Italian to conduct at least a basic telephone conversation and find out whether there's anything useful happening now. http://www.infolecce.it/chi-siamo/703.html?jjj=1628177847458

It seemed to me that most of the foreign tourists in Puglia were from the UK. If the UK folks haven't been traveling, that would have really reduced the demand for tours in the area.

Late last evening I saw some specific COVID protocols fior Puglia. Can’t remember exactly what, but I will research and post when I’m out of the Dentists chair in a bit.

In addition to any other COVID related entry requirements based on your Nationality or reason for visiting/studying, there is a self-report form to be filled out and sent to the first Regional Health Authority for the region you will be staying in.

Find the form here, as well as a list of health authorities and their email addresses:

https://www.sanita.puglia.it/web/pugliasalute/ingresso-in-puglia-da-estero

You could book 4 or 5 nights in Lecce at La Bella B and B if they have a vacancy. Roberto, who speaks excellent English (he studied in Dublin and is married to an Englishwoman) offers tours, as shown on this page.

https://www.labellalecce.it/index.php/en/

Lecce is on the train line south of Bari.

Here is another option if the dates work out:

https://www.hfholidays.co.uk/holidays-and-tours/guided-walking-altamura?format=pdf&vid=1986

The week starts August 28. You can book the tour without the flight from the UK and meet the group either at the Bari airport or at the hotel (they will book a private transfer for you).

Look at AHI Campus Abroad program in Polignano a Mare, Puglia. One week, stay in one hotel and do day trips in the region.

Tours, right now, are few and far between.

You could visit much of Puglia by a mixture of train and bus.

Don' skip Lecce. It's less than 70 miles from Alberobello and is a good base for train travel in the area. It's also a wonderful place to visit. I would suggest a side day trip, if you can find the time, to Gallipoli. You can get a train from Lecce.

My suggestion would be to plan a circle route from Bari. Are you also planning to fly out of Bari?

Thank you, everyone. for all these suggestions! We haven't ruled out the possibility of flying into Bari and out of Brindisi. It is tough finding overnight tours that are operating right now so using Lecce as a base for day trips is a great idea. Is Lecce a better base than Bari in terms of bus and train travel and general safety? I really appreciate eveyone's help. This forum is great!

I don't think one is better than the other, it is just a matter which places you want to see and of looking at the schedules and determining how much time you want to be in a bus, etc. I would choose north or south myself, but we all have different preferences and efficiency levels (not a morning person ). A lot closes up in Puglia from noon to early evening, so you have to plan day trips carefully. Nothing can replace doing the legwork, so check those schedules so you will know what the reality is!
It also depends on your arrival and departure times and whether it is four nights (three full days) or five nights (four full days)--that can make a big difference in how much time you actually have.

I came across this company that specializes in Puglia on another travel website. Have not used them, but might be worth checking out.

https://www.arrivedercipuglia.com/index.php/

Here is another I found:

http://www.experiencepuglia.com/

Friends of mine have used InItaly.com for some very successful trips.

Thank you so much for these suggestions! The weather in Italy has been hot. And many of the suggested tours seem to be operating in September/October.

Not sure when you're going, so this might be too late for your trip. If not then check out puglialy.com they do small group tours (7 people or less) and day trips. I have booked an one week tour with Pietro for next spring so I haven't personally taken his tour. But I've been very pleased with his quick responses to my emails and his reviews on TripAdvisor are excellent.

Thanks so much for this suggestion. We've had to put off Puglia but are still hoping to get there in Sept/Oct.