Southern Italy and Sicily

If possible fly into Naples if you don’t intend to visit Rome

Fast train from Rome to Naples then local commuter train to Sorrento
( or hire a driver)

There are many many threads here discussing the Sorrento/Amalfi area and transportation issues
Spend some time seaching the forum-filter by date and type
Then come back with specific questions

A good guidebook will also be helpful

Lots of good info here
https://www.sorrentoinsider.com/

And here under individual towns
https://www.ricksteves.com/europe/italy

The train schedules can be found on the Trenitalia and Italotren websites.
Be sure to do a day by day plan, like
Land Rome x days
Train to x, x days
Train to X, x days
You may find it gets tight if you need to get back to Rome for outbound flight. You could consider going to one of your other destinations first and working your way back to Rome.
Read up on the differences between Naples and Sorrento to determine which is right for you.

I would recommend taking the train from Rome to Salerno and either a ferry or bus to Amalfi. Amalfi is a transportation hub and it’s easy to visit coastal towns such as Ravello or Positano. You can the travel to Sorrento by ferry or bus and then to Naples from which you could fly to Catania or Palermo. It’s faster to get to Pompeii and Vesuvius from Naples and the Archeological Museum in Naples is terrific with many artifacts from Pompeii. For a phenomenal view of the Bay of Naples, the city, Isle of Capri and Vesuvius, take a funicular to Castel Sant’Elmo.

You'll have 14 full days in Italy. By the time you see all you want in the Naples/Amalfi area, your remaining time is going to be inadequate for a comprehensive look at Sicily. I recommend taking a good look at Rick's guide to Sicily to figure out which sights are most important to you. Sicily is a large island with important sights scattered all over. There are no fast trains, and plenty of places only have bus service (but probably none on Sundays). If you're willing to rent a car for part of your time in Sicily, you will be a more efficient traveler. But you still won't have enough time to see the entire island.

Going through Rome means you fly to Rome FCO, then take a train to Rome Termini, then a train to Naples. If the flight avails/prices force you to do that, but I would try to fly direct to my destination if I could.

You can fly direct Newark to Palermo. Spend 6-7 days exploring Palermo and western Sicily. Then take the overnight ferry to Naples and wake up arriving in Naples harbor without wasting a day traveling. Grab breakfast, hit some Naples sights as they open, (since you're early you be able to store your bags for free at the Archeological Museum) then train to Sorrento/Salerno which is your base for the rest of your Pompeii and Amalfi coast time. If you're in Sorrento, you can take the handy bus from Sorrento right to the Naples airport for your connections home (choose a later flight). If not, you can check connections from other Amalfi Coast towns.

You're on a tight schedule so check to see how connected Amalfi is to Naples. I honestly don't know. Sorrento and Salerno are probably your best bets connection-wise.

You mention Vesuvius. Are you thinking of trekking up the slope? Just bear in mind, that it's very weather-dependent so be flexible with your days.

Just my opinion: A tour that combines both Pompeii and Vesuvius on the same day is IMO, might be selling you something you don't want. Pompeii is a full day of walking. It'll be warm even in late September so I don't know how much energy you'll have to trek up Vesuvius. It's not a long hike, but after tromping all over Pompeii, I would have had zero interest in walking up those steps from the Vesuvius lot. And if they shortchange Pompeii, I think they're doing you a real disservice. Everybody is different so YMMV.