Flying to Sicily

Alitalia is defunct, it went out of business October 2021. Its successor is a smaller airline that retains a lot of Alitalia planes, and is called ITA. If you take ITA for both legs, your baggage, if you check it, will be checked through to Palermo--Punta Raisi airport. Ryanair might be cheaper for the Roma-Palermo leg, but you'll have to collect your bags and then board the Ryanair flight. It's up to you whether that's a bother or not. The NYC-Roma-Palermo route is the usual way to get to Palermo, but you can also go through Milano's Malpensa airport.

I flew to Milan (MXP), then after a couple of days, flew to Palermo (PMO) via easyjet After traveling around Sicily, I flew from Catania back to Milan. before flying back home.

Do consider flying into Palermo & out of Catania, to avoid backtracking.
This is a good choice, but only if it works with your itinerary, of course.

Wishing you a wonderful trip!

P.S.
Do book your tickets in advance for discounted prices.
If traveling carry-on, consider getting the, ‘upfront or extra legroom seats’, which allow you to bring a personal bag (under seat) plus one carry-on luggage. Check here for more details: easyjet.com/baggage

Catania is a somewhat larger airport than Palermo, but either will work.

RyanAir is, as far as I can tell, the most problematic of the budget airlines in that it works hard at finding ways to penalize you. Heaven help you if you decide to use a printed boarding pass, lose it on the way to the airport and have to reprint it. I think that costs something like $75(US). So far I've managed to avoid RyanAir completely; if it was my only choice, I'd use it; it's not unsafe.

Check carefully about luggage limits for checked as well as carry-on bags. Tickets that seem to be priced similarly may have very different costs by the time you add the baggage fees. And seat selection may well be extra also.

If you're flying onto Palermo right away, after landing in Rome, then I vote ITA booked all the way through if possible, they will be more likely willing to put you on a different flight if your connecting flight is delayed. I don't have first hand knowledge, but someone else mentioned they will also work with you if your flight to Rome is a different airline and is delayed. Family members last year missed their Ryan Air flight because of a delayed flight from NYC (by about an hour in the morning) and Ryan Air charged them full same-day price (i.e.$200/person) with no credit for missed flight.

The best route is whatever works for you. On our trip we flew to Catania via Munich and from Catania back to the states via Frankfurt. We didn’t want to fly through Rome.

Yes, check your options vis-a-vis Catania and Palermo. I had originally planned on flying into Palermo and out of Catania, but flipping the destinations actually saved me hundreds of dollars. I was not expecting that would make much difference.