Here are my suggestions for visiting Catania, Taormina, the cyclops riviera and Stromboli
Fly into Catania (Catania is a beautiful baroque city and definitely worth visiting)
You can rent a car from the Catania airport- tends to be very inexpensive
Catania is very close to the volcano Etna, the area around Etna is very famous for wines with a very high mineral content. I recommend Tenuta San Michele- a lot of fun and they serve a wonderful lunch. Must reserve in advance
Acti Castello and Aci Trezza- small fishing villages 30 min from Catania that are lovely to visit. These towns are part of the cyclops Riviera because legend has it that this is where the cyclops threw rocks at Ulysses http://www.touring-italy.net/tours/tour-details.php?recordid=18
Taormina- I would recommend making this home base for then visiting Catania, Etna, Aci Trezza, ecc- you can either stay in the hilltop town or down below the town near Isola Bella (a beautiful little beach just below the town). Taormina has a Greek amphitheater overlooking the sea (check their website for potential concerts http://www.teatrogrecotaormina.com/concerts/events.html or you can just visit the theatre itself during the day). Make sure to get a drink at the very chic hotel Grand Hotel Timeo, on their terrace overlooking the sea. Also make sure to get a daily granita from bam bar.
Visiting the Aeolian Islands. From Taormina you will drive north about 1 hour to Milazzo which is the port that has many daily departures for the islands. You can choose which island you would like to visit but I highly recommend Stromboli- very small with no cars and the most active volcano in the world. You can hike up the volcano with a tour guide.
The boats from Milazzo to the islands are notorious for changing times published online or based on the weather. Here is the website for tickets http://www.ferriesonline.com/?_ga=2.99046302.1258665796.1499163797-230175527.1499163797
If you have time on your trip I recommend that you visit  Villa Romana del Casale in the town Piazza Armerina. It is 80 min inland from Catania but features the best Roman mosaics I have ever seen (and I lived in Rome for 7 years). It is a UNESCO world heritage site. My favorite mosaic there? The women in bikinis doing different type of sports in the gym. http://www.villaromanadelcasale.it/