On our second trip to Sicily,we stayed in Taormina, picked up a rental car from Auto Europe, delivered to our hotel for a small charge, and then drove and explored the South coast. If you rent a car in Taormina, remember that there may not be any parking at your hotel, and you may have to keep the car at a lot on the edge of town. Our Hotel had a small parking area.
We did the Northern side (Palermo, on a seperate trip), but I jut consulted my Lonely Planet travel guide ( I don't go to Sicily without carrying it), and they said a bus trip between Catania and Palermo is only 2 1/2 hours. I would not be afraid of driving between Taormina and Palermo,just be aware that driving in Palermo is not for the faint-of-heart, especially during rush hour. We parked right downtown near the water when we were there, and had no problems getting in or out. You would take freeways most of the way, the A18 to Catania, and then the A19 across the island right to Palermo. Both are wonderful roads.