It would be helpful if you let us know your travel plans.
Getting from the airport
It's just over an hour to drive from Catania airport (Aeroporto Fontanarossa) to San Michele di Ganzaria and a couple of hours from Palermo. From Catania, it's pretty well signed from the airport for Gela or Caltagirone (SS417). Continue on this road past the turning for Caltagirone Sud (you don't want to go Caltagirone), then it will shortly be signed for San Michele di Ganzaria (SS124).
It's possible to get a bus direct to San Michele di Ganzaria from Catania airport, although there aren't that many buses - and none on Sunday. Cartagirone is better conected by bus to Catania.
I haven't managed to directly link to the timetables for San Michele or Cartagirone, but you can find them by clicking on the Catania area timetables, then click on "MIRABELLA IMBACCARI - San Michele di ganzaria - CALTAGIRONE - CATANIA". Unfortunately, the row for buses to San Michele often shows up as a question mark or square. We believe the 13:45 and 18:00 from "Aeroporto Fontanarossa" go to San Michele without needing to change in Caltagirone - but the other buses probably do not. You can buy the ticket on the bus.
As for the coach that you will need to catch from Catania airport, I believe it will leave from the airport at about 5.30 to 6.00pm. It's not very punctual...You have got to turn right and walk for a few hundred metres once you come out of the airport and you'll find a bus stop. A lot of coaches that all look the same will go past around that time. The coach may be blue or white. As coaches approach the stop watch carefully for the sign: Caltagirone, San Michele, Piazza Armerina, Mirabella. You need to get your ticket once on the coach and I would ensure you say to the busdriver: Per San Michele? I often do that myself, just in case....Also, don't get off in the first town the bus stops, that is Caltagirone. You want to get off at the next stop. You should be arriving in San Michele at about 8pm.
It's not possible to get directly from Palermo to San Michele by bus, although from Palermo centre you can reach Piazza Armerina which is about 17km away. There are buses to Piazza Armerina to San Michele on the Catania area timetables then click on "CALTAGIRONE -San Michele di Ganzaria - PIAZZA ARMERINA".
AST has many buses all over Sicily - details are on their website (Linee e orari = lines and timetables).
Flying to Sicily
The nearest airport is Catania, which is directly connected to the UK by Air Malta (from Gatwick) or BA (Gatwick). Alitalia fly from Heathrow, but you have to change in Italy. (With Alitalia it's worth making sure you have quite a long stop in Italy, as Laura has found they often don't put your luggage on the 2nd plane if the connection is tight.)
You can fly to Palermo direct from Stansted (with Ryanair) or Gatwick (Easyjet) but this means a longer trip in Sicily.
There are also some charter flights, but inconveniently only at weekends. Thompson flies to Catania from Gatwick and Manchester, but only on Sundays, until 25th October. Monarch flies from Manchester to Catania on Saturdays.
Unfortunately from other airports it isn't so easy to reach Sicily. As far as I know you'll need to change planes, and some routes involve changing airports in Milan or Rome.
For cheaper travel, and more adventure, you could get a budget flight to Rome (or Naples or ...) then take the train down. We've both done this, it's a beautiful -but long- train journey and if you time it right you can also see a bit of Rome.