The fastest way is a 1-hour flight. The most romantic is the train that crosses the Strait of Messina on a boat. Sicily is an island — all routes involve either flying over water or crossing it.
Ryanair, Wizz Air, ITA Airways, easyJet all compete. One of Italy's busiest domestic routes, prices start at €19–29. Catania airport is 15 min from center (Alibus €1). Palermo airport 35 min (bus €7).
The legendary route. At Villa San Giovanni, the entire train rolls onto a ferry and crosses the Strait. The night train departs Rome ~9:30pm, arrives Palermo 8am — couchettes from €45.
Parts of the A3 in Calabria are still two-lane with construction. Car ferries from Villa San Giovanni to Messina run every 20 min (€35–42, 20 min crossing).
Fly. Unless it's your first trip and you want the night train experience (genuinely wonderful), the 1-hour flight for €25 beats everything.