As said above, it really depends on the total contours of your trip. Where else are you going besides Palermo? Even for a just-Sicily itinerary, you may do better with using Catania airport (which is actually larger, with more flights) for at least one leg.
If Sicily is part of a larger Italy trip, there are flights to Palermo from many Italian cities (Rome has by far the most, but if you're already in Italy, there's no need to detour to Rome to get to Sicily).
If you want to go right to Sicily from the US, start by seeing what shows up in Kayak. Consider open jaws (multi-city) - into one city and out of another. If you get an itinerary all on one ticket, you are "protected"; the airline is responsible for getting you to Palermo in case of a delay or other problem on one of the legs.
If you are going to fly to Rome on one ticket and then from Rome to Sicily on a separate ticket, consider doing what I did (I was using miles to get to Rome, but couldn't use them to get to Sicily). Buy two tickets from Rome to Sicily, one for a few hours after your scheduled landing in Rome, one towards the end of that day. You will use one ticket and throw away the other, but it's still cheaper than a last minute ticket if you miss your connection.
If you are using Alitalia for domestic flights within Italy, look at their "International" website as well as their US one. Sometimes prices are lower on the International one, or deals are available on that site that are not sold on the US site. And, particularly if you are checking a bag, consider Alitalia; by the time you add fees charged by the budget carriers, they may not be any cheaper.