If you land in Catania, you can choose among various shuttle services which provide connections to and from the Aeolian Islands. You have to get a bus/shuttle/taxi to the port of Milazzo, which takes approx. one hour and half. Between the busses and shuttles you can opt for: Alibrando (shared shuttle service, € 25 per person, if the plane is late they automatically put you on the next shuttle departing); Giuntabus (€ 30 a/r public bus service); Eolian Shuttle (shared shuttle with ferry ticket combined). The boat companies to reach the islands are: Liberty Lines; Siremar; NGI.
During Summer (mid June – mid September) there is a ferry leaving from the port of Palermo. Liberty Lines departs at 14.00 from Palermo and at 7.45 from Lipari and it takes 4 hours.
During Winter you must move with the train station of Palermo to Milazzo train station, then get a taxi /bus to the port and then the ferry to Lipari.
Landing in Reggio Calabria is pretty straightforward: you take a taxi to the port (the ride is about 10 minutes) and there are a couple of ferries departing for Lipari. The boat ride takes 2 hours.