Flights to and from Sicily seem to be sparse, slightly erratic, and variable in price.
I doubt there is any direct flight from US, so you would land somewhere in EU, and go through border control. Of course you should check, but I would bet that a flight from the US to Palermo with a layover will be much more expensive than a roundtrip to Milan or Rome, then buying a secondary roundtrip to Palermo.
You can also fly to Palermo or Catania from Rome or Milan. These flights are cheap and frequent - many each day - and only take an hour or so. Although they might leave from a different airport. For example, I flew to Malpensa (Milan) then flew to Catania from Linarte (Milan) which would probably take a 90 minutes to be on the safe side to change airports.
I believe there is an overnight ferry from the port near Rome to Palermo too. I know for a fact there is one from Naples (1 hour south of Rome). You leave out 9 pm and arrive in Palermo at sunrise - either a private room with beds and private bath, or save money sleeping on the "theatre" seats.
The Catania airport is only a 4eu bus ride of 10 minutes into the city. The Palermo airport is essentially a few cities over and is a slid 45 minute train costing about 12 eu if I recall (could be wrong though). You can also arrive in Catania and leave from Palermo or vice versa. I think it takes about 6 hours by train to get between the two of them - although you pass some wonderful places to stop such as Taormina and Cefalu.