travel from taormina to matera

By air. Bus or train from Taormina to Catania, bus to airport, fly Catania to Bari, regional train from Bari to Matera.

Bus and/or train/bus combination (plus ferry across the Straits of Messina) is over ten hours. There is a night train but it's not direct.

You could rent a car and drive to Matera without having to go through Bari.

Check Rome2Rio.com to see transportation alternatives with links to schedules.

Just to add to Zoe's reply that there are buses from Taormina right to Catania airport, eliminating one step.

Thanks, Hsrold, I forgot about that airport bus.