To reach the Aeolian Islands, and in particular the embarkation of Lipari, the possibilities are different:
booking a flight to Catania or Palermo (tickets can be purchased on the internet with the main low-cost airlines).
Once in Catania, on board a shuttle bus you will have to reach the Port of Milazzo or Messina. For Milazzo there are various bus or private shuttle options, including www.eoliebooking.com/shuttle/ and https://giuntabus.com/milazzo-aeroporto-catania/ . To Messina take the bus that runs 5 times a day (www.saisautolinee.it).
From Milazzo and Messina you can reach Lipari by hydrofoil: www.libertylines.it .
Or from Palermo airport take the bus that leaves every half hour (www.prestiaecomande.it) and takes you directly to the Port of Palermo where you will board the hydrofoil that reaches Lipari in about 4 hours (www.libertylines.it).
from Rome you can take a night train with bunk that leads directly to Milazzo (www.trenitalia.com/).
From 8 July 2021 a connection was introduced with Italo that leads to Sapri (province of Salerno) and then by hydrofoil to Lipari (http:///www.italotreno.it/it/destinazioni-orari/messina-e-isole-eolie). Train and hydrofoil tickets can be purchased directly on Italo.
from Naples you can take the hydrofoil that leaves every day at 14.30, directly to Lipari, from the Port of Mergellina (NA) with the Company SNAV (www.snav.it). In 5 hours of navigation you will arrive at your destination.
Alternatively you can choose a night ferry departing from Naples at 20.00 and arrive in Lipari the following morning (www.carontetourist.it/it/siremar).
from Salerno you can take the hydrofoil Alicost that reaches the Aeolian Islands in 4 hours. (https://www.alicost.it/aliscafi-per-isole-eolie/).
get to Reggio Calabria airport, take a taxi to the port (10 minutes ride) and then hydrofoil to Lipari with Liberty Lines (www.libertylines.it).
Or take hydrofoil to Milazzo or Messina (www.libertylines.it) and then train/hydrofoil to Vibo Valentia (www.trenitalia.com/ o www.traghettilines.it/it/isole-eolie/traghetti-reggio-calabria-lipari.aspx )