I lived in Sicily for four years, in a small town close to Catania, and loved it.
Must-see places are Taormina and the ancient city of Siracusa. As you travel from Catania to Taormina, the seaside resorts of Acitrezza and Acireale are a nice stop along the way. The catacombs of Catania are worth a visit as well. Stroll along Via 20 Settembre for some nice shopping, or check out the side streets for antique shops. Take a drive up Mt. Etna, see the small town of Nicolosi.
Caltagirone, where gorgeous tiles are made, is a must-see. There is even a tiled staircase in the midst of town! Please, please budget to take home some of these beauties!
Visit the ancient temples of Agrigento.
There is so much history in Sicily, such good food and wine, wonderful warm-hearted people.
Food-wise... you'll be so delighted. A seafood salad with seafood so fresh and delicious, dressed only with a little olive oil and lemon juice. Grilled swordfish steak with patatine fritte. Tiny, grilled lamb cutlets. Girarrosto is also very good - chicken cooked on a spit, with a quartered potato inside it, a sprig of rosemary and a clove of garlic. Perfection! Try caponata as a first course... and cannoli for dessert.
I'm really spoiled when it comes to food in Sicily, the home-made kind. My landlady cooked from scratch not once, but twice a day. Small portions, but always fresh. A pasta, followed by vegetables and little meat or fish, followed by fruit. Ripe, juicy, delicious. Her MIL still baked bread daily, the soft, flat loaves.
I wish you a memorable, wonderful trip! In case you come through the tiny town of Motta St. Anastasia on your travels, give a wave to my Sicilian home!