Palermo to Naples by train

I wouldn't do it because the journey is too long (train has to be loaded onto ferry to get to Sicily). You can fly the short distance in an hour or so - fares are very inexpensive via budget carriers. You can take public transport within Sicily and around Naples instead of trying to connect the two (I think that's a better idea).

The weather in Sicily and southern Italy will almost certainly be brutally hot next summer. Believe me, I know; I did Sicily and Puglia back-to-back in 2015. It's one thing to be in heat like that for a very few days, but day after day for two weeks is draining.

Aside from weather issues, I would not recommend that, with a maximum of 2 weeks, you try to see both part of Sicily and part of mainland Italy. Sicily alone begs for more than two weeks even if you have a rental car at your disposal. I did not, and I found I had to leave out a lot of desirable places despite having 18 days for Sicily alone.

I would not recommend a rail pass. It will almost certainly cost you more money than you would spend on individual tickets. As you dig more into guide books, I suspect you'll find that at least a few of the places you want to go will require use of buses; Procida, at least, will mean a ferry. The rail pass won't help you there. Furthermore, it sounds as if you have an interest in some less-often-visited places and will probably not be covering a lot of miles. That will make your individual train tickets relatively inexpensive. If you end up with one or two long travel legs (such as Palermo to Naples) that include an InterCity or Freccia train (no Frecce on that specific route, I don't think), you can consider buying a non-refundable/non-changeable ticket when those initially go on sale, saving a lot of money.

Have you investigated transatlantic airfare yet? A lot of people find Sicily expensive to fly to from the US and end up at least considering a much less expensive flight to a distant airport, plus a budget flight on a European carrier. There are risks involved in doing that, but if the savings are really huge, many will take those risks. Obviously, a rail pass would not help you with the plane ticket from your landing point down to Sicily.

I just looked at ferry schedules for winter for another thread. There is a night crossing from Naples, leaving just after 8 pm and getting into Palermo a little before 7 am. A 2-berth cabin is probably less money than a room for the night + air fare.

From rome2rio.com, it looks like there is train service from Palermo to Cefalu to Taormina to Catania to Siracusa. Then fly to Naples from Catania. That would surely fill up your time and all great places to visit.

We are considering the opposite direction for next year, not in summer but either in April or Oct/Nov--Naples by train to a little coastal town in Basilicata called Maratea (~2.5 hrs by Freccia), staying there 2 nights as it's supposed to have world-class scenery & appears pretty much unspoiled by tourism. From there, rough timing to Taormina, Sicily--with a change in Paola, the train/ferry to Messina and a change there--is about 6 hours. Stay in Taormina--2nd time there--for 2 nights, then train back to Messina, change to Cefalu, that journey taking ~3.5 hours. 3-4 nights there, fly out of Palermo. Early lines on train costs are very inexpensive, especially bought in advance.

I know it doesn't do justice to even half of the island but I'd rather have some time than none. And we have a soft spot for Taormina as it was our first destination on our first European journey eight years ago.

Again, I don't know if I'd do this in hot weather, but I'll try anything once. Oh, acraven mentioned transatlantic airfares. We've flown Lufthansa and are happy with them, and I know that they have a Palermo/Munich flight, which is a hub to go almost anyplace from there.

Enjoy your planning!