You can travel to the island of Santorini in two ways: by ferry boat or by plane.
1. By ferry boat The central port of Santorini is called Athinios and is connected with various ports of continental Greece as well as ports of other Greek islands.
Both high speed and regular ferry boats depart on a regular basis from the port of Piraeus. The trip with a high speed ferry takes about 4 hours, while a regular ferry takes about 8 hours to reach Santorini. High speed ferry boats are depart from the port of Rafina, taking about 6 hours to arrive to Santorini.
Ferries also arrive to Santorini from various islands of the Aegean, from Crete, as well as from the Dodecanese islands. Some of those islands are Mykonos, Folegandros, Ios, Naxos, Paros, Milos, Kimolos, Serifos, Sifnos, Tinos, Syros, Kos, Karpathos, Amorgos and Koufonisia.
2. By plane
The National Airport of Santorini (JTR) is located 6km away from Fira, the capital town of the island. Flights to this airport arrive from Athens, Thessaloniki, Heraklion in Crete and Larnaka, as well as from many countries such as France, Austria, Italy, United Kingdom, Germany, Serbia, Switzerland, Finland, Denmark, Norway, Katar, Spain and Hungary.
Useful phone numbers:
Port of Piraeus: +30 210 414-7800
Port of Rafina: +30 22940 288888 and +30 22940 22300
International Airport of Athens "Eleftherios Venizelos": +30 210 353-1000
Port of Athinios (Santorini): +30 22860 22239
As for your transportation while on the island, there are many buses that offer access to all towns and villages of the island (Telephone: +30 22860-23812). Alternatively, there are taxis available 24/7 (Telephone: +30 22860-22555).
In Fira, there is an aerial tramway available, which you can use to descend to the bay of Mesa Yalos where you can take the boat that takes you on a tour of caldera (Telephone: +30 22860-22977, +30 22860-23045). If you want to discover the underwater part of the caldera, a tourist submarine is also available for a tour that lasts about 1 hour.
Lastly, if you would like to experience the most traditional way of transportation, you can rent a donkey in Fira for about 3 euros.