There is no airport on Ithaca. The two practical routes in are by ferry from neighbouring Kefalonia and by long-distance ferry from the mainland.
From Kefalonia: the short crossing from Sami to either Pisaetos or Vathy takes 30–60 minutes and runs several times a day in season. Kefalonia has an airport with direct seasonal flights from many European cities, and is the obvious arrival point if you are flying in.
From the mainland: ferries run from Patras (around 4–5 hours) and from Astakos in Aetolia (around 2 hours), the latter being the cheaper and quicker option if you are driving down from Athens. Schedules vary by season; check current Ionian Group or Levante Ferries timetables.
Once on the island, a hired car or scooter is essentially required to see anything beyond Vathy. Local buses exist but are infrequent. Distances are short — nowhere on Ithaca is more than 45 minutes' drive from anywhere else — but the roads are winding and steep.