The island of Ko Phangan... The place where it all started. It is probably easiest to fly into Phuket and then visit a local tourist bureau there who should be able to hook you up with a ticket to the island.
Ko Samui is definitely worth some time while you are there too.
One thing to remember is that although there is a great deal of fun to be had at these Full Moon parties, a lot of people get robbed, mugged, or otherwise - Keep your friends around you and your stuff locked up.
http://fullmoonparty-thailand.com/ about 8 years ago
The full moon party is on Koh Phangan, at the south of the Island at Haad Rin. It's probably nicer to stay somewhere else on the island though where it's more secluded and quiet and get a tuk tuk down to Haad Rin and back each night if you'd like a bit of quiet each morning while you recover and if you'd like to be able to go in the water when you go to the beach. The beaches at Haad Rin are very dirty because of the Full Moon Parties.. I wouldn't swim in the water there. The rest of the island is also a lot prettier.
I stayed here and it was great; http://www.sawadee.com/hotel/kohphangan/goldenhill/
I recommend visiting Koh Tao as well while you're there, it's a very pretty little island just to the north of Koh Phangan.
PS If you're tempted to go to Mushy Mountain at the full moon party, it might be a better idea to wait until the night afterwards when it's quieter, and stick to alcohol in buckets for the full moon party and the nights leading up to it. about 8 years ago