Southern Italy

It's too much for me--heck just Pompeii and Herculaneum in one day would be too much for me (and I am only 45)--but maybe not too much for everyone. How were you going to travel to Paestum and then Matera?

Matera is worth seeing, but not just for one night with presumably a lot of travel time on both ends.

We have a driver that will take us from Sorrento to Paestum and then on to Matera

So that is doable then, but you'll have an evening and morning in Matera and then have to head to Bari for your flight. I would save Matera for a future trip, but if you feel like that is not in the cards and have the stamina it's not crazy.

Trying to see both Herculaneum and Pompeii in the same day is sensory overload! Add to the fact there is not luggage storage at Herculaneum.

Edited
From their site.
ATTENTION PLEASE!
We inform all visitors that, in order to protect the frescoes and the environments of the archaeological areas of competence of this Superintendence and in order to guarantee safety of everybody, is STRICTLY FORBIDDEN to introduce inside the excavations of Pompeii, Herculaneum, Oplontis, Stabia and the Museum of Boscoreale, handbags, backpacks, luggage, cases, larger than 30x30x15 cm.
Visitors can leave their bags and personal belongings at the free locker room at the access gates.
However, during some periods the sites are very crowded and the locker room may not contain the baggage of all visitors, it appeals to the sense of cooperation of each person to ensure that the new measures can be fully applicated.

Pompeii does enable you to store your bags on site, but try to hit here at opening time as there is little shade and can be sweltering.

Even though you'll have a driver, your ride to Paestum will take at least an hour from Sorrento, IF your driver heads up thru the Lattari mountains. Visiting Paestum deserves at least an hour, which would be very rushed. You then want to take a 3+ hour drive to Matera. Spend the night. The next day, maybe wander a bit around town then head to Bari over an hour away. Then fly to Rome.

It's your choice, but please rethink your itinerary.

So it sounds like I don't have enough time to do Matera. Would 3 nights in Sorrento and 2 in Positano sound about right for the activities we would like to do. We plan on spending 1 day in Capri. Any other recs while we are in this area? Thanks

I didn't mean to sound harsh, but seeing it on paper does put it into perspective.
I do think you'll have a much more enjoyable experience by staying on the coast.

So, if we stay in Sorrento 3 nights should we stay in Positano or Amalfi the next 2 nights?

Are you arriving Naples after an international flight, or coming from somewhere in Europe?
This makes a difference as to what is actually feasible your first day.

I would stay on the A/C for the one day. Matera was wonderful but you really need more than one day to enjoy it.

I think visiting Herculaneum and Pompeii in the same day, with an early start, is doable, I did. However, I suggest you do that by train from Sorrento, without having any baggage with you. Much less hassle. From Naples to Sorrento, a nice splurge is to take a taxi or car service direct from your lodging in Naples to your lodging in Sorrento. Cost would be about 100 euros, but would save you considerable time and hassle. I did this, had a half day and then lunch in Naples, got picked up at 2 pm, relaxed during the drive, arrived fresh at my lodging in Sorrento by 3:15 pm, then was out for the late afternoon, dinner and evening in Sorrento.