The nearest airports are Catania and Palermo Airports (60 km and 260 km away). From Catania Airport you can get on the Interbus bus to Taormina, everyday from 7 am to 7 pm (it usually takes 70 minutes).
All trains coming from northern Italy, from Milan or Rome, whose destinations are Syracuse, Catania or Agrigento, stop at Taormina - Giardini. Here you'll find a taxi service (15 minutes), otherwise you can get on an Interbus bus to Castelmola (20 minutes).
Autostrada A18 (Messina-Catania) exit to Taormina. Go to Taormina city centre and follow the road signs to Castelmola (15 minutes away).