7 day Amalfi coast stay

You can see by looking at a map that Amalfi is far from ideal as a base to reach Pompeii or Paestum.

You may need to think about what priorities are and them possibly consider having 2 bases while in the AC area.

You go nowhere fast on the AC road. A rental car is usually not 'the answer' for transportation needs, when visiting the AC

Where/when do you fly in/out? What month?
I felt like the walk between Amalfi and Atrani was only about 15 minutes, so I think that would make a fine base for visiting Positano as well as the other AC towns. Try to mostly use ferry out of Amalfi--buses get crowded.
I don't know of any reason to day trip to Sorrento. Paestum would be a very time consuming day trip, but I would consider using the driver for that, also visiting a bufala mozzarella farm. Pompeii is best visited en route back to Naples -- presuming your flight is out of Naples, you would need to spend the last night there anyway, and you could see Pompeii on the way (either driver or public transit/ storing bags).

Vickie, I suggest you investigate (perhaps with the Search Box top center) how relatively difficult it is to reach the Amalfi Coast, even from Sorrento or Salerno (which haven't been mentioned.) There is no doubt you have probably found a lovely and quiet place to stay, in a beautiful environment. But (aside from everything you have mentioned) you need to plan NOW for your last day and your transfer to Pugli (which has limited transportation as well.) Even though you're not going to the Rome airport, it's going to take a long time to get there - just like it is going to take a long time to get to Pompeii or Capri from the villa.

I'm not telling you not to take the villa. I'm telling you not to make rash assumptions about First World accessibility. Did you mention the month? Things can be seasonal down there, like ferry schedules. We did not stay on the AC, but rather in touristy Sorrento, because we knew we wanted a daytrip every day, and it has the best transit for such trips. (5 nights for us.) Clearly, you plan a more leisurely visit than ours. We spent the first night in Naples because even though we only came from London, the plane was late enough that it was hard to get to Sorrento after dinner. Just for comparison, we paid 240 Euros for a car and driver, one day, to see Positano, Amalfi, Ravello, and two view points, in an eight hour excursion from Sorrento.

Will you have to walk 15-minutes to find a restaurant? Is the villa stocked with food? You will be better situated for Paestum than we were in Sorrento. We discarded the idea of going to Paestum. You should download a SITA bus schedule for the month you're going to be there.

Also, assuming you are driving to Puglia, I would pick up the car in Salerno, and then you could see Paestum that day.

You can easily be car free. I'm assuming your flying into Naples Capodicino airport.

Take the Taxi, with 4 of you the taxi is probably best, to Napoli Centrale.
From there, take the 40-50 minute train ride to Salerno. At Salerno use either the ferry or bus to Atrani. Bus/ferry schedules. https://www.positano.com

My preference is to use the ferries when at all possible. However the busses will not be busy this time of year.

You have plenty of time to see everything on your list, including Pompeii Scavi. Use this tour,
https://www.positano.com/en/c/sunland-travel/i/SJPY8.
Or see if your Naples driver can combine the two. Do Pompeii first.

Traveling to Ravello, Minori, Maiori, Vietri Sul Mar is probably easiest by bus. You'll see much more. On the day you do the southern towns, purchase the 24 hour bus pass. It will save time as well as $$.

To reach Sorrento and Capri, use the ferry.

Your going to have a great time.

Rent your car in Salerno as your leaving for Puliga. It will save time.

Amalfi is kind of the major transport hub on the Amalfi Coast, so you’re well positioned to day trip to many of the places you mention. Ferries are ideal, but there’s also buses heading everywhere out of Amalfi. You’ll mostly be day tripping against the tourist groups coming from the big cities. Take advantage of the early mornings and evenings when the crowds are minimal. You have a lot of flexibility with travel from here, so that’s good. Keep your options open.

I’m very curious about this villa! I stayed in Atrani and it’s a 15 minute walk between the two, so if it’s where I’m picturing it should be a great view.

Atrani has some great restaurants, and is much more authentic feeling than Amalfi. Ravello is a real highlight, and easy for you to visit from Amalfi. Positano and Capri are both easily accessible by ferry and worthy day trips. Paestum, though a little bit farther, is also a doable day trip. Ferry to Salerno and train to Paestum.

There’s no reason to see Sorrento over someplace else, unless you’re staying there. It’s a good base for visiting Naples/Pompeii however. If you’re interested in seeing either or both (you should) on your own, then I’d suggest spending a couple nights in Sorrento. Day tripping these places from Amalfi is tough. However, if you’ve got the money to have a private driver make this happen then by all means. This is definitely an area where a little cash splurge can reap real benefits.

Thank you to everyone who has replied--very helpful! For this trip, we'll fly into Rome, then home from Bari. I'm now thinking maybe we should have two bases for the first of our two weeks: a hotel or B&B in Naples for two nights to see that city and take an excursion down to Pompeii, then we'll have five nights in the villa between Amalfi town and Atrani, then when we leave, we'll pick up a rental car in Salerno and visit Paestum before going to Puglia. The trip will be in late April. I assume all the ferries will be fully operational then but if anyone knows otherwise, please let me know. I assume 5 nights will be enough to visit Positano and all the small towns and take a day trip to Capri, since we don't have to fit Naples, Pompeii or Paestum in there. In fact, would four nights be sufficient? Because if we can steal a day from this half of the vacation, it gives us an extra day in Puglia, and I'm finding that's a big area that's going to require multiple bases for all we want to see and do!

Vickie, that plan sounds perfect to me. Enjoy your trip!

Vickie, there are multiple ferry companies, so you either have to depend on local aggregating websites, or try to download multiple schedules, looking for the service DATES. Because the ferries to Capri from Salerno start up in April, you need exact 2020 dates, but for now you'll have to settle for the 2019 dates. Even in Sorrento (where there are more ferries with longer seasons) the companies do not cooperate or run rigorously updated professional websites. And you can't depend on a dude at a window to send you to the competition because they are leaving two hours sooner!

Your revised plan sounds great. For Puglia, Matera and Monopoli are both about 45 minutes from Bari airport if you want to try to arrange it where you don't have to do the "last night near the airport" thing.
Enjoy!

Four nights would be plenty, a day for Amalfi, a day for Ravello, though they could be combined. Day for Positano and a day for the southern coast.
Doing this would allow you to ferry from Naples to Capri. You can catch an earlier ferry from there. Then ferry to Amalfi from Capri. Just be sure to check the last ferry time to Amalfi. Or spend the night and leave for Amalfi the next morning.