Welcome to Sicily Aarcee54 – Grazie!
Five days is a limited time, but enough to whet your appetite!
Fly Air Italy from Naples to Catania – Intra-Italian flight are fast and inexpensive.
There’s no reason to fly from Palermo – Arrive and depart from Catania. Your areas of interest are basically located in Eastern Sicily.
When short on time, you would appreciate hiring a car, however it would depend on your base(s) and you could potentially accomplish your itinerary on public transport, or even with excursions.
If you were to decide on one base, Taormina would be your optimal base because you would enjoy many tours and group bus excursions, even to Agrigento’s Valley of the Temples and Siracusa, and Taormina is a convenient base for exploring Mount Etna.
http://www.satgroup.it/excursion_en.asp?cod=summer
However, Taormina is a tourist resort town – You might split your time between Taormina and Siracusa. Siracusa is an ancient town, and a 'real' Sicilian city with a mix of tourism and authentic Sicilian life. Both towns would be possible via public transport.
But if you love food, wine, culture, history, you might still consider another option with Ragusa-Modica, since the UNESCO Val di Noto is stunning, and this area enjoys some of the best cuisine and wines in Sicily-Italy.
http://whc.unesco.org/en/list/1024
This itinerary could be done on public transport, but with your limited time, if you were to decide on a 3-base itinerary, hire a car.
This earlier thread discusses some of your options.
Italy To Tromboli and Sicily:
http://www.tripadvisor.com/ShowTopic-g187886-i343-k4589527-Italy_To_Tromboli_and_Sicily-Sicily.html
Best Wishes!
Vagabonda