Getting there

Air:
Ryanair flies non stop from Trapani (Birgi) to Britain, Spain, Ireland, Sweden and Germany.
To the Palermo airport Falcone e Borsellino there are non-stop flights from almost every major city in Europe and even from New York.

Land:
At the moment there are three direct bus connections from Palermo Airport to Trapani. Departures are 11pm (Mon-Sat), 15pm an 20:30pm. From the Palermo Central Station (Stazione Centrale FS) there are buses every half hour. See the bus schedule here (click "Ricerca Orari").

Alternatively you can go Palermo-Trapani by train. This takes a bit longer than going by bus.

Car Rental
If you want a car on sicily book it prior to your journey. That is cheaper. Compact cars you can rent from Sixti in Palermo. Bigger ones from autoeurope, Budget, Sixt, et cetera.

Sea:
There are two ferry lines operating the route from Trapani to the Egadi Islands: The state-run Siremar and the private UsticaLines. Both dash down the tyrrhenian sea with "aliscafi"-called hydrofoils. Furthermore there are ferries to Sardinia, and Pantelleria and Tunis at least once a week.