Solo Traveling from Rome to Puglia, best way to spend 5 days?

What has looked appealing to you so far? It's a big area so a big ask to narrow it down. Basically, you can take train to Bari and explore that vicinity--Itria valley and immediate coast, or train to Lecce and explore Salento.

https://www.viaggiareinpuglia.it/

A week is too short to see everything in Puglia, but with a car you can hit a few good spots.

On my trip to Puglia last fall I flew in to Bari from London and flew out of Brindisi to Trieste. Pick the airport that offers the best flights schedule and price wise to your next destination. Both airports were small but fine.

That’s such a short amount by the time you arrive in Puglia! With essentially four days, I would take the train to Lecce, rent a car, and explore the Salento region during the days for the smaller towns & beaches.

Check your dates for any festivals or music events. I’ve attended some music concerts & the San Nicola Sagestra - great events!

I’ve stayed in the Puglia region for 10 days each the last two months of May. This time for you will give you the desire for a lot more to explore! : )

I greatly appreciate all of the advice!
To specify a bit more, I plan to be in the Puglia area for around 5 days on my own. Looking for recommendations on towns and beaches to visit that have a more classic Italian feel to them.

I very much liked Lecce and Martina Franca. Alberobello is so unique — you should try to see the trulli. I used Ostuni as a base but I was on a small group (limited to 10 people) 6 day tour and all transportation was provided for all of our daily excursions. Matera is unique too. Polignano al Mare and Monopoli are attractive seafront towns. It is a great region to visit Without a car, look for bus and train routes. Your transportation options may ultimately determine which towns you will visit. Also look into organized day tours that may include several towns.

I wouldn’t go all the way down to Lecce with just five days. Lots to see in Puglia — here’s one idea:

My favorite town is Trani, a little north of Bari. It’s a lovely little town with a seaport and picturesque streets. Drive or take a train from Rome. I’d spend 2 nights in Trani.

Then drive to Matera (1 night). Explore the lovely Val d’Itria to see Alberobello, Martina Franca and stay in Ostuni (1 night). Then perhaps Polignano a Mare (1 night) for a beach, and then a short drive up to Bari for your flight home.

Edited to add: there are some beaches in Trani but we found the best soft, sandy beach in nearby Margherita di Savoia, a very picturesque little town which we visited to tour the salt plains (producing more than 5 million tons of salt a year).

Looking for recommendations on towns and beaches to visit that have a
more classic Italian feel to them.

Do you want to stay in town and day trip to beaches or stay in a beach town? I liked staying in Lecce and traveling to each coast during the day. I seek out natural beaches with no services. By "classic Italian feel," do you mean organized beaches with chairs/umbrellas?
The beaches north of Gallipoli are spectacular if you want sand. For rocky beaches, I would look near Otranto.

You might look at the Southern Italy Rick Steve's tour itinerary to get some ideas. We did that tour last spring and loved Matera, where the 2 canyons in the middle of town have caves people lived in till mid-20th century. World heritage site, lots and lots of steps/stairs. People are friendly, food and wine outstanding. We stayed at Locanda di San Martino in a cave room. Had heat/AC and modern plumbing and simply beautiful. If you go to Matera and want a simple, delicious meal, I recommend Uacciardidd at Via Ascanio Persio 33, a main pedestrian street in Matera. Deli/butcher shop that offers variety of food and wine and they'll heat food up for you to enjoy at outdoor tables or take away. Wonderful owner/manager and his wife does the cooking. If you want fancy, Baccante restaurant is Michelin starred. Alberobello is where the trulli are, the cone shaped structures. Lots of tourists but charming just the same.