Which Town

Taormina is wonderful, but it is way over on the east coast and close only to Mount Etna. If you stay in Palermo you can get to many sights: Erice, Segesta, Monreale. All quite close. You don't want to miss Agrigento and Selinunte (if you're interested in Greek ruins, of course), both on the south coast, and the Roman villa at Piazza Armerina is terrific. It's sort of in the middle. We had a car when we went, so we went around the edge of Sicily (with a detour to the villa) and ended up near Palermo. The autostrade are great and very fast; the mountain and unimproved coast roads are amazingly slow. Have a great trip!

You have enough time to use two even three different towns as bases. Palermo should be one. It has much of interest itself; and you can easily do a number of day trips from there, including Cefalu and Erice. Taormina is lovely, but small. It could be another base. Two days would be enough there, with a third for Siracusa. I strongly recommend that you include Agrigento, either as a daytrip from Palermo, or as another place for a night or two. A word of caution. The distances in Sicily look short, but it may take more time than you think to traverse them.

I also suggest splitting your time between two or even three "hubs". Palermo, Taormina and Siracusa would be my choices. There's a small town next to Taormina with great ruins and a museum (Naxos) that has a number of B&Bs. We stayed at Hotel Continental in Taormina, beautiful terrace views. If you have a car, there is a public garage very nearby. The hotel itself is a staircase or two away from the center of town. In Siracusa, the island of Ortygia is the oldest part with the most atmosphere. Hotel Posta is on a main street and there are excellent bus connections into town (and a little free bus that goes around Ortygia and then to the train/bus station across the bridge). There's also a public parking garage near the hotel. I would put Cefalu on the list if you'd rather not stay in Palermo (although Palermo is more convenient for trips to Monreale and Erice). Cefalu has a large sandy beach, beautiful old town center and the cathedral has one of the most beautiful mosaics in the world.