Puglia this summer advice needed

3 nights Lecce
2 nights Bari
4 nights Martina Franca
2 nights Matera
2 nights Naples

Have you scoped out the transport yet? This may not be the ideal order, but the public transport in this region is what made me tap out and rent a car, so I don't know the details--someone here will. I was thinking it might reduce your travel time to do Matera first or last, and Bari first or last. Those are the two with best transit options to Naples.
You could also visit Bari as a day trip (or in transit, as you would most likely pass through). I like the old town, but it is small. If you do that, I would suggest one additional night in Lecce, the other in Naples if you have not spent time there before.
Look for places with pools to be sure!

Your Puglia locations are all close to one another, and four nights in Martina seems like a lot - it's not a big place - 50,000 people. Further north and on the coast is Trani which people recommend frequently. Also Puglia is not hard with a car and it will give you options in the heat. I talked with a cafe owner in Matera who said temperatures in summer hit 40C (104F) with some regularity - that's not weather you can just ignore.

As Mike mentions, it could be very very hot. Make sure your accommodation has got air conditioning

Thank you very much for your kind and useful replies. Sorry it took me so long to thank your messages.

Using your info and reading more I am thinking of making several changes .
As a reminder I share that we will spend 18 nights in south Italy from July 29 to August 16.

Let’s see, what do you think of this new itinerary?

Arriving in Naples ,
Sorrento, 2 nights.

Salerno, 3 nights.

Trani, 3 nights.

Monopoly or Polignano a Mare, 2 nights.

Lecce, 4 nights.

Matera, 2 nights.

Naples, 2 nights.
Flight home.

We would prefer to do this using public transportation.

I am very interested in visiting Paestum. Is it doable in the middle of summer time?
Am I choosing bases that are good for day trips?
Would it be better for logistic reasons to stay in Monopoly instead of Polignano or viceversa?

I am starting to feel the pressure of summer coming very soon and would love to book my lodgings very soon.

I appreciate very much all the help you can offer!
Thank you!

As regards the Puglia section, yes, Trani, Monopoli and Polignano are all quite close on the trainline. Considering it's going to be very hot and busy in August I'd be inclined to choose one base for five nights. Once your luggage is stowed it's easy to hop on an air-conditioned train to daytrip wherever you want by train along the east coast. Not so much fun dragging it around in the heat and waiting for afternoon checkin time......(a day wasted?) Since you're not interested in beaches I'd consider Bari for the excellent transportation links.....
https://slowtravelitalyspain.blogspot.com/2019/11/puglia-bari-in-april-trip-report.html
Also, Lecce is rather the outlier, Lecce to Matera by public transport will be long.......

PS you need to sort out your public transport links and then book ASAP because August is peak season

I agree with @Tinac that Trani and the other two are close together, I'd stay in 1 for 5 nites.

I think the early part of the journey is rushed, you may not have taken into account the time between stops. For example you arrive in Naples (don't say when) then need to get to Sorrento. Depending on when your flight arrives it may be the afternoon or evening before you're actually out and about in Sorrento which leaves just one full day. Sorrento to Salerno by public transit is 2+ hours so when all is said and done it will likely be the afternoon before you're out and about. Salerno to Trani is 6+ hour so most of the day will be gone.

Thank you very much for your useful replies!

We have decided to do what was suggested and will have 1 five nites stay and one four nites in Puglia.

We are supposed to arrive at noon to Naples and we’ll go directly to Sorrento. We have been to Sorrento a few times . We just need to share a hug with a good friend we have there.
I will be asking for further advice , thanks in advance.
Every piece of information you share with me has been highly useful!
Thank you!

Paestum is VERY NEAT, but I can think of maybe.....4-6 trees there? there will be NO SHADE, except in the shadows of the temples, so I would suggest going as early as possible on the (I THINK possibly hourly-ish train in the morning before a big break for the afternoon?) train from Salerno. Buy your train tickets both at the same time (from the tabacchi in the station) and make sure you know the schedules on the way back as the Paestum station is bare bones. They have a nice small museum too, which you can do when it gets hotter. Many folks pair a Paestum visit with a mozzarella di bufala farm down there, that might be fun for you! There is a Paestum "town" and beaches that many locals frequent, if you want to explore that too (but just did a street view walk around and it doesn't look terribly interesting except for sandy beaches). I normally take a bus to Vietri on my Paestum day after I return to Salerno, just for a little walk around and ceramics shopping. If you have a whole other day in Salerno, you can possibly take a ferry to any of the other tiny towns on the coast like Cetara, Maiori/Minori, etc (I'm not going to suggest a bus as one of my hottest days ever was on a bus with no air conditioning, although it was a Salerno city bus) or visit Castello Arechi and the Duomo.