Sorrento and southern Italy Itinerary

I'm confused by the timeline--you need more than a week--but it sounds like a great plan.
Salerno is very convenient--bus, train, and ferry all converge there, and driving out of the city is not a daunting proposition like Naples. If you are going to Paestum, I would just pick up the car in Salerno.
Perhaps you could drive across to Matera first, then head south to Longobucco, then loop back north along the coast (Cilento area is very nice). Are you an outdoorsy family? Calabria has an abundance of national parks.
If you manage to fit in Puglia, I love the Itria Valley where Alberobello is located. Some say don't stay there because of the day trippers, but the town is pleasant and if you stay there, you get mornings and evenings. There are several pleasant villages all close together, so I always say just pick based on accommodations.

Tight but it can be done- Suggestion Only
Fast train from Rome to Naples- Circumvesuviana train (lower level of the Main train station) to Sorrento. 1/2 day travel - 2.5 days to explore the Amalfi coast ( use the bus to Positano (2nd stop) 1/2 day then on to Amalfi 1/2 day. Take the ferry back to Sorrento - the view of the coast line is amazing.
Rent a car in Salerno - you may have to take another bus straight to Salerno or the Ferry . It is a 3.5 hour drive to Longobucco. Spend the night or 2
Travel next day to Matera 2.75 hrs. spend the day and night. I would advise a tour guide of the area as the history is so unique.
Travel next day to Alberobello 1.15 hrs and spend a 1/2 day and then 1 hour to Bari to return the car. You can fly out of Bari to Rome to catch the plane home.
I have left out Capri but you may want to spend the week just travelling to Longobucco and the Amalfi.

a bit off topic but since your wife's grandparents immigrated from the town there may be relatives or people who knew them. Amazon recently suggested a book to me "Sauce on Sunday" https://www.amazon.com/Sauce-Sunday-Ancestral-Journey-Sicilian-ebook/dp/B07KXWZYSJ which I haven't read but in the free preview "look inside" on Amazon the author mentions a few steps before her trip to get in touch with relatives, and I've seen other stories of people connecting with relatives. Probably she's already doing something along those lines, but I thought I'd mention it.

Thank you ever so much for all the input so far. Did not dawn on me to fly out of Bali to Rome in order to return to U.S. which makes so much more sense than what I was thinking and I am now going to build that in. As far as timeline, we are looking at leaving Rome morning of May 13th and so far flying back home on May 23rd which gives us a good 10 days for this adventure and has us not flying on a weekend ($$). Just two people traveling this leg and yes we are outdoorsy, not holding my breath but going to try and fit in the Path of The Gods hike.
Looks like getting a car out of Salerno is being suggested instead of taking a train, just need to determine best way to get to Salerno from Sorrento without duplicating what we may have already done while seeing the Amalfi Coast.

Will definitely take the advise of arranging a tour guide for Matera.
And Oh My, I will look into the book being suggested. My grandparents immigrated from Sicily so I think we both have some relatives to try and identify, how fun that would be.

Ten days is much better! If you are not visiting Naples or Pompeii, then I would bypass Sorrento and just head straight to the Amalfi Coast, or Capri--three days is the minimum I would devote to one or the other. You can also day trip to Capri from Amalfi. You could also stay a couple nights on Capri, which everyone says is superior to a day trip. But then half your time is gone.
You could certainly take a train further south and pick up a car, but that would make visiting Paestum more complicated. The town you mentioned might be tough to find a rental agency in that would do a one-way rental. You should look at larger towns that would have the larger companies.
I would want at least a couple nights in Matera, but you could see it en route if you get time-challenged.
Check Brindisi for return flights also, or Lamezia Terme if you go to Calabria last.
I'd honestly choose either Campania and Calabria or Campania and Puglia for this trip--trying to fit in Puglia too seems a stretch to me, but I move at a slower pace and realize some move very quickly. (Just for a point of reference, I could spend ten days in the Naples-coast area or in Puglia). Once you get a rough sketch and can see the drive times, it will probably get much clearer.

Robert, two years ago my sister-in-law and I went on the South of Italy tour. Afterwards we drove to Mongrassano Calabria where her grandparents were from. Your wife will love seeing her family history. I want to share my car rental research and driving experience with you. I'm not going to comment on the rest of your journey other than to say Alberobello and Matera are fantastic. Mongrassano is on the west side of Cosenza. Longobucco is on the east. Mongrassano is so small that it doesn't have a hotel. We stayed overnight in San Marco Argentano, about 3 miles away on a curvey road.

Rental car research: Our tour ended in Naples. Trying to avoid driving from Naples to Mongrassano, I looked at taking the train to Paolo. Changed that idea fairly quickly. The rental car company I contacted was an unaffiliated small rental company and it was the only one of the three with an email address. They wanted to charge by the mile plus the rental fee. I wasn't comfortable with that arrangement. There was also a timing issue in that they were closed on Sunday when I needed to return the car.

Next idea was to take the train from Naples to Salerno, rent a car there, return it there, train back to Naples in time to make our train to Venice Monday morning. Sounded great on paper and the rates through Auto Europe/Hertz were acceptable. However, the office closed at noon on Sunday which would have given us no time Sunday morning in Mongrassano. Returning on Monday was not an option timewise for us.

We ended up renting from Auto Europe/Hertz at the Naples Airport on Saturday morning and returning it there on Sunday late afternoon. The drive to Calabria was very easy. The highway is better than most in AZ and CA but once off the highway the roads are 2 lane or sometimes 1. If you are using google maps, Waze or ?app for directions you need to add at least an hour or more to your drive time for potty breaks, getting lost and food.

So my advise is 1) do careful research about the rental company especially the times they are open at your pick-up and drop-off locations. If you are renting or returning on a weekend or in a small town, you could run into timing problems. 2) have change handy for toll roads and use the cash lanes. 3) map out your drive and if using a GSP app, be sure your navigator can use it. My SIL tried and overall did OK but I had to pull to the side of the road several times to see the map because she "lost it" on the phone screen. We also got very lost returning to Naples because we missed the turnoff and ended up almost at the shipyards instead of the airport. Luckily I filled up with gas 30 or so miles south of Naples. The rental return person took pity on me and didn't charge us for not having a full tank of gas when I told him there was no way I was going to leave the parking lot to get gas, having figure out how to get to the opposite side of the freeway and back.

Oh, another lesson I learned is don't forget to update your credit card information with the car rental company if the card changes. My Visa was compromised and replaced. I contacted the hotels with the change but forgot the car rental.

Have a wonderful trip.
Kathy

Don't know if this helps, but my wife and I are doing southern Italy early fall this year, and after flying into Naples and seeing the Amalfi coast area (without a car), my plan is to take ferry from Amalfi to Salerno. Pick up rental car there. Then use car to see southern Italy stops (Matera, Alberobello, Ostuni, Lecce, etc). Then we are dropping off car in Bari and flying out from there (I think you could do the same thing from Brindisi)

I would pick up the car in Sorrento. That saves a commuter train back to Naples (there is no south-bound train departing Sorrento) or a bus or ferry to Salerno. Driving out of Sorrento is easy enough if you stick to the largest inland roads and try to avoid rush hour traffic.

You can easily see Alberobello en-route and feel that you got the idea of it. But if you plan to fall in love with it or take a lot of photos, and you have the time to spare, then you might want to stay overnight.

This was the first posting I did and what a treasure of great advise that I have incorporated into our agenda. Thank you all for your responses. Except for the tail end I have my semi-rough itinerary outlined, added some days to the trip (13 full days) and have return flights on 14th day out of Bari, in process of looking at accommodations.
My question:
I notice the Rick Steves books do not cover Matera, Alberabello or that area of Italy? I will be driving a car that I picked up in Salerno after two nights in Sorrento and two nights in Amalfi Town. After spending two nights in Longobucco, will drive to Matera for two nights and then have three open nights (two days) before spending two nights in Bari (want a full day to see a little of Bari) before turning car in and catching flight home. Besides visiting Alberabello, I am looking at Locorotondo, Cisternino or Ostuni. Is there someplace else I should be investigating as well, or a beach town I can spend a day or afternoon relaxing at? Rick Steves makes it easy with his books but when not covered I am doing allot of web reading (which I enjoy).
Day 8: Full Day in Matera
Day 9: Leave Matera, travel to and see Alberabello, Stay?
Day 10: open
Day 11: open
Day 12: From somewhere travel to Bari and check into hotel
Day 13: Day in Bari
Day 14: Flight home (noon)

Lonely Planet covers the area, but the web should have you covered.
Near Alberobello you have all of the Itria Valley villages. They are all close together, so I would pick based on accommodations you like. I have stayed in Cisternino, which I enjoyed very much (even though I am a vegetarian and the town's specialty is grilled meat, in case that appeals). Close by at the coast you have Monopoli, where I have also stayed and adored--and Polignano a Mare. Both are not a long drive into the valley to see the trulli. There is also a cave complex you might be interested in touring.
Alberobello does fill up with tour buses during the day, but if you stay there you have mornings and evenings, and can be out exploring during the day, so I would not rule it out. The bakery I enjoyed the most on that trip was in Alberobello. Silly aside: Locorotondo is pleasant, but I had the worst sandwich of my life there, it sort of tainted it, lol. Noci is a cute tiny town with an excellent place for lunch if you pass through.

PS We did a tour of an abandoned village near Matera called Craco--very interesting!

I would stay in the historic center of Ostuni or Cisternino rather than in Alberobello. Possible to do a day trip to Lecce from either.

We found both Polignano a Mare and Alberobello interesting for a couple hours in the early morning or evening --- we visited them in March and in November, so heaven knows what they are like in the season when the buses of people arrive. For such a tiny town, Polignano a Mare has a HUGE parking lot, so that gives me some idea.

Giovinazzo looks good to me as a seaside town, but we haven't been there --- we enjoyed Trani very much as a place to stay with a harbor and fishing boats, but that's maybe too far to go.

We've stayed in a trullo twice for a week each time --- it is fun and there are plenty to choose from in the countryside.