There are private bus companies which have scheduled coach trips between various cities in Sicily and a well established bus network services each city. Alternatively you can use taxis where necessary.
You can reach Sicily through one of its 2 coastal airports, either Palermo or Catania. The Palermo airport is 31 km west of the city and modern airport buses run every 30 minutes to the city centre. Catania's airport is only 5 km south of the city centre and buses run into the city every 20 minutes till midnight.
If you happen to be in Southern Italy then you will need to use a train-ferry from Calabria to Messina's Stazione Centrale. From there you can continue your trip by train to any of the various coastal cities of Sicily.