From Catania airport is now easier to reach Taormina thanks to the new railway station, called just Catania Fontanarossa.
At the exit of the airport terminal there is in fact the stop of the shuttle bus (a bus of AMT Catania) that goes to the train station. The ticket costs € 5.60 one way. The schedules are available on the website and on the Trenitalia app. Tickets can be purchased directly at the station (ticket machines) or online.
As soon as you arrive at the Taormina-Giardini station, a red shuttle called TaorminaLink will connect you to the city center. All included in the price of a single ticket.
https://www.trenitalia.com/it/treni_regionali/sicilia/catania-aeroporto-fontanarossa.html