After many trips to Italy, finally, Sicily is on our list this spring! Will start in Palermo and train and bus in country thereafter. Will end up in Taormina, and will have to backtrack to Catania to catch a flight. I know about RyanAir and Vueling, any other local airlines I should check out?
Where are you flying to?
To learn who flies where from a particular airport, go to that airport's Wikipedia (a great trick I learned on this Forum!).
Here's the destination list for Palermo: https://en.wikipedia.org/wiki/Falcone%E2%80%93Borsellino_Airport#Airlines_and_destinations
and for Catania: https://en.wikipedia.org/wiki/Catania%E2%80%93Fontanarossa_Airport#Airlines_and_destinations
As you see, while Palermo is the larger city, Catania is the larger airport, with more flights and destinations.
For getting from Taormina to Catania airport, there is a direct bus; you don't need to go into the city of Catania unless you want to.
If you are looking at domestic Alitalia flights (say, between Sicily and the mainland), look at Alitalia's International website: https://www.alitalia.com/en_en/. This often has cheaper prices than the US website for these flights; just don't forget that it's a foreign transaction, so you may need to notify your credit card company and/or pay foreign transaction fees.
Also, you may find my VERY detailed trip report helpful. I was there in April-May 2014 with a friend, and we drove, but had a very similar itinerary to yours (start in Palermo, end in Taormina). https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina
Thank you so much, Harold! I will check out your itinerary for more helpful hints. Roberto, our plan is to fly into Palermo directly from the States.
If you're flying right into Palermo, first check the cost and routing available for flying from Brownsville to Palermo, and Catania to Brownsville, all on one ticket. To do this, use the "multi city" option on Kayak, Google Flights, etc.
If you don't like the price or routing, you can investigate the possibility of flying to another European city from the US, and using a separate ticket for a flight to Sicily. When you read my report, you'll see that's what I did. But note the precautions I took. For my arrival, I booked two different tickets from Rome to Palermo; one was 3 hours after scheduled arrival, and one was 10 hours after scheduled arrival. I knew I'd use one and throw away one, and counted the price of this leg as the price of the two tickets together (that was still less than one last minute ticket, as we were flying on a Monday). For my departure, I flew from Catania to Rome the night before my Rome to US flight, to be sure I wouldn't miss it.
Note that while various cities have flights to Palermo and Catania (the Wikipedia lists will show you which ones), Rome has, by far, the most flights per day, if you need flexibility.