The closest and most convenient airport to fly into for Taormina is Catania. Palermo is too far afield. Transportation to Taormina from the airport is as follows:
Local bus-The bus (Interbus) is € 5,10 (8,50 return), every hour and takes 1.10 hours (but there are a few slow buses as well).
The terminal in Catania is close to the train station in Via Archimede / Viale della Libertà but the ticket office is nearby in Viale della Libertà / Via D’Amico, the terminal in Taormina is in Via Luigi Pirandello 100 meter before the pedestrian Corso Umberto. It has also a stop at the Taormina-Giardini train station.
From Catania airport (when coming out of the airport turn right) buses take 1.25 hours and ere € 8,20.
Possible to hire cars quite reasonably from the airport if you so desire and utilize the car over the period of stay if needed for sight seeing i.e. ETNA. There is a large multi-story car park situated at the south end of Taormina as you enter the town off the main highway. Some Hotels in Taormina have access to their own parking so please check before you book.
Taxis-there are many taxi hire companies in Sicily and at Catania airport which you can pre-book before you arrive and they will wait for you in the arrival terminal. Prices vary form 60-70 euros upwards depending on the number of people travelling together and size of luggage. Some useful sites are as follows: