Where to go for 10 days in Italy
 

Two couples have talked about taking a trip to Italy for years! We think we're ready! We are 60-2 men & 2 women-love wine, good food and experiencing different cultures! Realizing we would love to go ALL over Italy, it is not possible in 10 days, so we are selecting the Amalfi area. We would love a nice home ($250-$500 nt) and make some day trips. We have been to Rome, but other than that, nowhere else in Italy. Please share your suggestions on areas to stay, day trips to make, places to eat, whether a trip to Capri is worth the time, and anything else you can share. Thanks so much from Dallas.

What time of year?

First, read up on the different locations and determine which ones you want to visit.--day trips (for example, Pompeii, Naples) and transport make all the difference in this region. Some people prefer to unpack once; others like to experience a place at night and avoid long day trips.
Based on what you have listed so far, I'd split the time between Positano, Capri, and Naples (last night before departure -- provided you are flying in/out of Naples). I think Capri is best experienced as an overnight.

My vote would be Sorrento. You'd be near the local train, the ferry, and a local bus. Check ferry stops and you'll see schedules for visiting Capri, Positano, town of Amalfi, etc.

Early/Mid October 2020

You will not regret going to Florence! It is packed with amazing churches, museums, and is lined with incredibly beautiful streets. Food is unforgettable. If you end up going to Florence, you have to try local steak!

Looks like there is not a Non Stop flight available from Texas to Naples, so I guess we'll fly into Rome and take the train to Naples. Thinking of staying 2 nights in Naples (do Pompeii) then take a taxi (that is the consensus online?) to the Amalfi Coast area. We would like a home, so we have read to look into Sorrento or Atrani. Any thoughts? We would like to visit Ravello, Salerno, Positano, Paestum, Amalfi..................any comments on other places, whether these are good choices and is there enough to see for a week in this area? Another question, after reading comments online, is whether Capri is worth a visit in October? If we wanted to cook fresh seafood a couple of nights, is there a good market to get fresh fish and vegetables?

It could look something like this:
Arrive Rome, take bus to Naples (there is a direct airport bus, Fiumicino Express, and I would prefer that, but otherwise head to Rome and then train to Naples).
Three nights Naples--one day for Naples itself, one day for Pompeii, then the next day head to Paestum (store luggage at Salerno), return to Salerno, ferry to Amalfi. If staying in Atrani, then I'd hire a driver.
About a week in Atrani or Amalfi (about a 20 minute walk apart). One day for Ravello, one day for Positano, one day for Salerno and Vietri, one day for Capri
Return to Rome via Salerno, last night before flight/