Because questions like yours come up on this board almost weekly, you may find the Search box top left, blue type, to deliver vast amounts of useful information. Most of your reasonable ideas have a large number of important details to learn. For example, approaching Pompeii Scavi (the ancient city) from the Sorrento side and from the Salerno side involve two entirely different train companies, and different distances from the train stations to the Scavi. A Rome-Naples-Pompeii-Sorrento trip requires a change in Naples to the local commuter (unreserved) line, Circumvesuviana, which is especially hot and crowded. It sounds like you may already know that Pompeii Scavi entrances have very reliable luggage storage.
This area will already be hot and crowded by the last week in May, so air conditioning, and patience, are called for. There is only one two-lane road along most of the AC (don't use the word "Amalfi" to mean Amalfi Coast because it's also the name of a town), and it moves very slowly. A private boat tour will be much more expensive than ferries. You will find that taxis are quite expensive, because this area has high tourism costs. Can all your luggage fit in one sedan trunk?
Your travel style is your own, but I would not change hotels as often as you are planning to. We spent five nights in Sorrento (last week of May), which is a purpose-built, postwar resort town, because it has the best transportation options. But that meant we only saw the Amalfi Coast from an eight-hour car and driver excursion to Positano, Amalfi, and Ravello. Positano was a must-see for me, but I never need to go there again. Maiori and Minori are more accessible from Salerno than from Sorrento. From Sorrento, we had to write off the hope of seeing Paestum, because it was just too far. Edit: We enjoyed our daytrip to Capri, but we skipped the time-sucking Blue Grotto.
https://community.ricksteves.com/travel-forum/italy/amalfi-coast-for-7-nights-itinerary-help
https://community.ricksteves.com/travel-forum/trip-reports/trip-report-2-weeks-of-day-trips-from-sorrento