The quickest way is to fly. Daily flights connect Lampedusa with Palermo in about 35 minutes, with onward connections to other Italian cities. In summer, direct charter flights run from Rome, Milan, and other Italian cities. The airport is right on the edge of town — you can walk to most places from it.
If you prefer the sea, the SIREMAR ferry from Porto Empedocle near Agrigento takes about eight hours overnight, stopping at Linosa along the way. Book in advance if you want a cabin or plan to bring a car — summer sailings fill up fast.