JFK to Sicily via Rome-where are Customs?

You will go thru Immigration in Rome -passport check as your entry to Schengen

Customs in Palermo which is basically just a walk thru the door unless you have something to declare

Customs, bringing goods into a country, is usually just walking through a door that says “nothing to declare.”
When returning to the US, you will declare anything you bought over $800 each and pay a tax.

Most Americans call immigraton and customs......"Customs." But there is difference as was pointed out above.

So the answer to your questions is customs will be done in Palermo. But your passport will be checked in Rome.

Trying to figure out timelines.

Are you planning on separate tickets or just trying to figure out when you will walk out the door from the airport in Palermo?

Thank you all for the replies. The flights are arranged by the tour we are on. We have free time after our arrival and I am just trying to figure out how much that might be-against afternoon closing times for things we might try see. Yes, here in the USA, we regard immigration and the sometimes separated declaration of goods as one entity,Customs. If Immigration is built into the time in Rome before our second flight and Customs in Palermo is essentially a walk-through, then can allow more time in the afternoon. Pending, of course, airlines, politics,weather, Covid,etc.

On second thought, and not that I don't believe it, but this makes no sense. I could bring in contraband,pass through immigration in Rome, pass it on inside the terminal to a confederate and then have nothing to declare in Palermo? Unless we don't see our luggage until the end?

You won't see your luggage until Palermo (Make sure when you check in at JFK it's labeled to Palermo.)