There are plenty of options to arrive at Lipari that you can choose from.
From Catania Airport: the shortest trip to reach the Island is taking a coach just outside the terminal from a company called SAIS Autolinee. The coach will take you to Messina city where another coach connection will take you to Milazzo harbour. From Milazzo you can take an hydrofoil (1 hour) to Lipari from two different Companies: Liberty Lin. From Palermo Airport: As a family that will arrive from Palermo Airport, during the summer season, this is the best option for you and your family. Just outside the Terminal you will find a coach every 1/2 hour to Palermo Harbour where you will take an hydrofoil directly to Lipari Island. PS: make sure you book in advance your ticket, check the timetables and the price before you will take this journey to avoid stress and dissatisfaction. (PLEASE CHECK THEIR WEBSITE BEFORE TRAVELLING. For a cheapest option from Palermo Airport, you can take a coach from the terminal to Palermo Rail Station where you can take a train to Milazzo. From there, 1-hour hydrofoil service will take you to Lipari Island. There are many other options by airport. You can easily arrive from Lamezia Terme airport. At the arrival a local bus or taxi will take you to Lamezia Terme Rail Station where you can take a train to Reggio Calabria. During the summer season, there are hydrofoils that will take you directly to Lipari. PS: make sure you book in advance your ticket, check the timetables and the price before you will take this journey to avoid stress and dissatisfaction. (PLEASE CHECK THEIR WEBSITE BEFORE TRAVELLING. If you arrive by train, the closest Rail station is Milazzo. From there, you will take a local bus or a taxi to the harbour where you will find hydrofoils directly to Lipari Island. If you will come from Messina, there are hydrofoils connections directly to the Islands. By car, you will arrive from the freeway. Take the exit MIlazzo/ Isole Eolie at the freeway; follow the direction to the harbour where you will find ferry to Lipari Island.