We were in Sicily last summer and had a superb holiday. As the previous poster said, Sardinia, Pompeii and Malta are a bit too far to go if you're based in Taormina.
I would recommend heading south towards Siracusa (should be plenty of bus tours offering day trips in Taormina), well worth the visit (especially Ortigia, the old town). If you can stay overnight, then the Valley of the Temples along the south coast near Agrigento is also worth seeing.
But Etna is very accessible from Taormina and is a great day trip by bus. If you're fit, you can spend half the day trudging up and down the lava cones. A cable car can take you to the summit from La Sapienza (expensive though), very cold at the top. The recent eruptions at Etna could make things interesting. Try and organise a trip to a local vineyard, plenty of those around the slopes of Etna.
You're probably wise in not hiring a car if you're not a very confident driver, the Sicilians are insane as regards their driving behavior.