Hydra has no airport. The only way in is by sea: fast hydrofoil and catamaran services run several times daily from Athens' main port, Piraeus, taking about 1 hour 50 minutes. The boats are operated by Hellenic Seaways and Blue Star and tend to be heavily booked in July and August — book ahead online.
There is no car ferry; cars are not allowed on Hydra anyway, so leaving the vehicle at Piraeus or Metochi (a small port on the mainland 5 km opposite Hydra, with a short water-taxi service) is the only option. Once on the island, donkeys and your own feet are the transport; small water taxis run to the beaches and to Kamini and Vlychos if you don't fancy the walk.