If you land in Trapani, take a taxi or shuttle to the port and catch a Liberty Lines hydrofoil to Favignana.
If you land in Palermo, first take a bus or train to Trapani, then go to the port and board a Liberty Lines hydrofoil to Favignana.
Check schedules and buy tickets at www.libertylines.it.
You can view some touristic information using the following links.
TBA.
Some information on the visa requirements to Italy:
http://vistoperitalia.esteri.it/home/en.