By plane
The international airports of Palermo (Falcone/Borsellino) and Catania (Fontanarossa) are flied to by foreign airlines (for example Swiss, Lufthansa, Austrian Airlines) by scheduled and charter flights. Alitalia offers daily scheduled flights from several German cities, Zurich and Vienna via Milan respectively Rome to Sicily.
By train
There are no direct train-connections from foreign countries to Sicily. But there are direct night trains with sleeping cars and couchettes from Milan, Turin, Venice, Triest and Rome to Palermo, Syracuse and Agrigent. Reservation is highly recommended; all places in trains to Calabria are mostly taken. Bikes are best put into special bike-bags (also an advantage by train - ransportation within Sicily). You can with it prevent a lot of trouble and discussions. Further it’s cheaper this way.
By ship
From Genoa, Livorno and Naples, the car ferries of "Tirrenia" res. "Grandi Traghetti" run to Palermo, from Naples additionally the "SI.RE.MAR." via the Liparian Islands (Stromboli, Lipari etc.) to Sicily. It’s also recommended to book in time, because the seats (Poltrone) are very uncomfortable, especially by heavy sea.
By car
Who wants to travel there by his own car has to take upon himself a long and hard car trip. Furthermore he’ll have to pay several charges in Switzerland, Austria and Italy. It also is, to say it in a positive way, a real challenge to manage the traffic-jams in several cities.