Mt. Etna- I highly recommend you take a tour to Mt. Etna, the tallest active volcano in Europe, to see the volcanic craters. While I was there the volcano was smoking, which made it extra cool. Go prepared to hike in the cold. Bring a very warm coat, good hiking shoes, and gloves. I was not prepared for how high up it was, and ended up falling in my crappy shoes! I also was not prepared for how cold it was. Oops!
Museo Storico dello Sbarco- A unique museum about the Sicilians experience during WWII.
Museo del Cinema Catania- A fun kitschy museum that you have to see, and it's right next door to Museo Storico. They have designed rooms set up, and in each room they play movies that are only set in those rooms.
La Pescheria- A seafood market from 8am-2pm every day except Sundays behind Piazza del Duomo. It's fun to walk around and look at all of the fresh fish, some still alive, like the eels. There are a few stands with meat and cheese at this market too.
Savia- A great cafe to get arancini at. I had the arancino Catania (with eggplant), and the arancino ragu (with meat). Both were good, and the cafe is historical and cute.
Caffe Prestipino- A charming place to try delicious sweets with a cappuccino. I had the Minni di virgini a very sweet pastry in honor of St. Agatha, and a mini cannolo.
Agora Hostel- A great place to grab a glass of wine with welcoming service. In the wine cellar there is a lava cave.
Murgo Winery- The wine was good, and they are generous when you do the tasting. The cheese and charcuterie they serve with the wine is fantastic! Actually, the best I had in Catania, especially the pecorino and spicy salami. The view is wonderful over the vineyard and the prices are reasonable. They also have a store where you can purchase wine to bring with you. You would have to hire a driver to bring you here, or include it in a tour.
I recommend you stay in an apartment in Catania. There are many to choose from, and with so many markets and specialty wine shops it is nice to be able to bring food back with you and cook.
There is a great street, Via S. Michele, with cool shops on it. A record store, vintage store, local female artists store, and handmade leather good items store just to name a few.