3 days Rome to Pompeii and Amalfi Coast

You really have one full day and two half-days. Skipping Napoli means you will not see the archeological museum which contains the finds from Pompeii and other nearby cities.

You will have to change trains in Napoli Centrale ( you could check your bags at the luggage storage service and take the metro to the archeological museum) to go to Sorrento (Circumvesuviana station is downstairs); on the way you can stop to see Pompeii (there is a baggage deposit at the site). Then pick up your bags and get back on the Circumvesuviana to Sorrento. From there, you can take a bus to the Amalfi Coast and stay in Amalfi or further down the Coast (Vietri sul Mare is one of the furthest points, since you said you want to see all of it). Or stay in Praiano or Ravello but you will have some backtracking on your full day.

The next day, you can do a linear tour of the Coast, all going in one direction, by bus and ferry. I would take the bus to Amalfi, transfer to another bus for Ravello, have a few hours there (maybe lunch), then back to the ferry to go to Praiano and Positano. Return to your hotel that evening.

The next morning you will have an early start to make train connections to Mintepulciano; by bus or ferry return to Sorrento (or take the bus in the opposite direction to Salerno and take a direct train to Roma Termini from there, it will save you at least a train transfer and a flight of stairs with luggage (Circumvesuviana station at piazza Garibaldi/Napoli Centrale).

On such a short trip and in VERY hot weather, consider skipping Pompeii and seeing Herculaneum. It is smaller, can be seen in 2 hours, and better preseverd. Pompeii, while incredible, is vast, with long hot walks between exhibits, there is no shade, it is very exposed, personally, I would not visit there in summertime. The Naples archeological museum is a must to see what was taken from Pompeii and Herculaneum. Two hours at the museum, two hours at Herculaneum and then relax you are in Italy! :)

ckt, use the search box top center and read as many past threads on this topic as possible. You need to get an idea of how difficult access in and out of the A.C. is, and how crowded it is in July. You didn't say whether you have a rental car, but you might regret taking it to the A.C. in summer. There are zero trains between Sorrento and Salerno (that is, across "the full coast"). Much of the road route is a single, two-lane road, with curves so tricky that busses have to stop and back up to get around them. It is slow and crowded in the summer.

You need to allow a lot more time to get around once you reach Naples on the first train (?), and perhaps switch to the Circumvesuviana line to go to Pompeii. Then you'd continue to Sorrento and take the SITA bus (very crowded, might have to wait for the second bus ... ) to, say, Positano. This area books up months in advance. If you don't have rooms now, you may need to take your fifth or sixth choice.