You can reach Sicily by plane (last time we flew with Ryan Air from Torino to Palermo, which has a small but modern airport - Catania is the other one, together with the small one in Trapani), by boat (we took the ferry twice, once from Genova and once from Civitavecchia) by car (you will have to take the ferry called "Caronte" from Reggio Calabria to Messina -the ride is 20 minutes long and very conveniently organized) or by train (which also goes on the ferry to cross the strait).
To go to the Aeolian islands, you take the ferry either from Napoli in Campania, Villa San Giovanni in Calabria or Milazzo in Sicilia. While for the Aegadian the ferries leave from Trapani, and so are the ones leaving for Pantelleria. Pantelleria has also an airport with daily flights from many Italian cities, and so does Lampedusa.
We always rented cars in Sicily, last time with Hertz in Trapani, where they were super helpful! Driving can be a little challenging in the big cities because of the traffic and especially the amount of scooters ;-)