flights to Sicily from NYC

I'm assuming you plan to have a single ticket showing both flights in each direction, right?

Minimum connection time varies by airport. You can probably find it pretty easily by Googling something like "connection time AMS", "connection time FCO", etc. I think AMS may be as low as 50 minutes. But that is just the minimum required for issuing a ticket, not a guarantee that the timing will be OK for the specific flights you end up using on the exact day you are traveling, and in larger airports the minimum can vary depending on which terminals you are flying into and out of. Be sure you find the precise information that would apply to your situation.

Another key data point is how many later flights are available on the second airline (or perhaps a partner airline) from your connecting point to your final destination. Your first flight could be significantly delayed; we all know that happens often enough. If you're on a single ticket, your airline will be responsible for getting you to your final destination. The question is: Will it be able to do so relatively quickly? The more onward flights available, the likelier that some empty seats can be found for your use.

I like to have at least one hour more than the minimum connection time for the airport where I'm connecting and the specific terminals I'll have to transit. I will choose a different transfer point to avoid a tight connection if there's not too much difference in the fare. At some level of fare difference, I'd take a chance on the tight connection, but I'd be preparing myself mentally to miss the connection and be very delayed. For me, that would be much worse going to Europe (sleep deprivation + possibly a missed vacation day) than returning home.

I've noticed some comments here, both positive and negative, about changing planes in some specific European airports. I vaguely recall (I hope folks will correct me if I'm misremembering) that AMS and FRA are considered pretty easy transfer points, whereas CDG and LHR are tougher. But that ease-of-transfer should already be factored into the minimum connection time.

I'm going to let others address your question about customs and immigration since I'm always too zombified when I arrive in Europe to remember what happens at each airport. I do know that if you're on a single ticket, your checked baggage will be transferred for you; you won't see it until you reach your final destination.

Both PMO and CTA are served primarily by European low cost airlines.

The only traditional airline that serves extensively both Sicilian airports is Alitalia, which is a partner of Delta. There are numerous Alitalia flights connecting both PMO and CTA to Rome FCO.

Lufthansa also connects the two to Munich (MUC) but with only one daily flight, therefore you may not be able to complete your trip in one day and may be forced to spend a night in Munich in both directions.

I don't believe KLM flies to either Sicilian airport from AMS, maybe they start next summer, but I haven't checked. Certainly not currently.

Given the above, I think your best option is to fly to Sicily via FCO with Alitalia (or a combination of Delta and Alitalia, on a code share ticket).

If your stopover airports are either Rome or Amsterdam you will go through immigration as these are where you enter the Schengen area.

You then catch your connecting flight to Sicily where you collect your checked luggage. You then pass through customs in Sicily, which is a non event (unless you are declaring goods).

Phyllis, in 2014 I flew from JFK to Palermo on a United ticket. First flight was overnight JFK to Munich on Lufthansa, connecting to a 2-hour flight from Munich to Palermo on Dolomiti, which is a Lufthansa regional affiliate. All went well. I did not return directly, but instead flew Volotea from Palermo to Naples, train to Verona and Venice, and flew back to the US from Venice.

Thank you all for your help!