We will be flying into Catania. We would like to spend our first day or two relaxing and getting acclimated to the time difference at a charming fishing village not far from the airport. We would very much appreciate recommendations for the village
No specific recommendation, but the airport is just south of Catania, so wherever you stay will either be north of the city, or south of the nature reserve that is just south of the airport. In this case, not far will probably be a minimum of 30-40 minutes away.
What is your next destination after arrival? I’d go there directly.
There isn’t much near the airport, which is right next to Catania near the industrial district of Fontanarossa.
Aci Trezza, part of Aci Castello, north of the city, is a nice fishing town made famous by the Italian novelist Giovanni Verga, however it’s on the opposite side of Catania coming from the airport.
"What is your next destination after arrival? I’d go there directly."
I would as well. Note that from Catania airport, you can take direct buses to many places in eastern Sicily, so you don't even have to go into Catania itself (of course, some places will have better connections if you do go via Catania).
Though it’s not a little fishing village, if it’s not on your itinerary, consider the island of Ortigia in Siracusa.