Filicudi can be reached easily from Milazzo (20 minutes from Messina), and from Palermo.
The nearest airports are Palermo, Catania and Reggio Calabria.
Usually from outside italy it is easier to get to Palermo Airport in the morning, then take an hydrofoil (at around 2pm)from Palermo port to Filicudi.
From Milazzo you can take a ferry ( Siremar, Ustica Lines, NGI), or an hydrofoil.
In the links section on the right there are useful links to trasportation companies like : Siremar, Ustica Lines, NGI, Sais, Giuntabus.
By air you can get also to the town of Catania, there take a bus to messina (SAIS), where you'll easily spot the GIUNTABUS bus to Milazzo near the arrival bus station