Southern Italy and Sicily-Train or Rent a car?

Puglia and Matera can be done (though with some effort) but I would wager Calabria will be considerably easier with a car.
If you share your specific Puglia locations, we can better advise.

I plan on establishing a base in Lecce and exploring Ostuni, Otranto, Alberobello and then going to Matera.

Can you recommend a nice hotel in Napoli? We stayed near the Teatro di San Carlo in 2018 and enjoyed the area.

Thanks,

I plan on establishing a base in Lecce and exploring Ostuni, Otranto,
Alberobello and then going to Matera.

Be sure to look at travel times. It is difficult to say at this point of itinerary whether car would be better or not (I decided yes on the car).
You mentioned hiking as well--some of the Murgia Materana park can be reached on foot from Matera, though of course if you want to get to farther reaches you would want a car. Most of the hikes we did in Puglia required a car.

Can you recommend a nice hotel in Napoli? We stayed near the Teatro di
San Carlo in 2018 and enjoyed the area.
I liked Hotel Piazza Bellini.

Ultimately, we need more detail and the plan to be fully fleshed out before transport logistics. I think it's a lot for three weeks--we need to know the Sicily plans and how much time you want to spend in Naples (and if you will include the Naples area.)

Without a car, covering that ground in three weeks will require significant and painful compromises. I don't know anything about Calabria, but Sicily is a time-consuming beast of an island. The trains, where they exist, are slow. The buses mostly don't run very frequently. You can visit Sicily without a car (I did), but I think you need about 3 weeks to do so if you want to see all the highlights. My 17-18 days were not enough, and I had no interest in (and therefore skipped) the major Greek sites at Segesta, Selinunte and Agrigento.

Train travel within Puglia can be slow and tedious, so I might consider renting a car in Lecce to access the other parts of the region. I can't really speak to Sicily, as we've only visited on a cruise.

Thank you everyone for your thoughtful feedback.

When I went to Sicily in 2018, we did rent a car, and I thought that was the right thing to do. We visited Siricusa, Catania, Taormina, Palermo (revisit Monreale), Cefalù, Agrigento. This 2025 trip (our honeymoon), I would like to go to Trapani, visit relatives in Terrasini, Mt Etna, spend more time in Palermo, Piazza Armerina.

We wish to spend three or four days in Napoli, and take a day trip to Pompei. I love Napoli. I studied there in 1990 and the two days we were there in 2018 were inadequate.

You all have me convinced--rent a car and stay in hotels that have parking.

My next question is what is the best route to take the car from the mainland to Sicily, via Messina? Is that very expensive, does anyone know?

I don't know anything about the crossing to Sicily, but I know others here have experience and will chime in. Looking at your Naples and Sicily plans, I think you should limit it to that and make stops in Calabria en route but save Matera and Puglia for the next trip. That would be my three-week plan anyway---good luck!