Koh Phangan is only reachable by boat from Samui, Koh Tao or the mainland. The closest large city on the mainland is Surat Thani. Boats from Surat Thani usually stop at Samui as well. The trip can take from two to three hours.
The closest airport is on Samui. From the Samui airport, it's just a short ride to the pier where boats leave for Phangan. You can buy a combined car and boat transfer from the transportation counter in the airport. The trip from Samui to Phangan takes just 30 minute or so.
Lomprayah offers a combined bus and boat service from Bangkok to Koh Phangan, and on to Koh Tao if desired. Buses leave Bangkok twice daily, stop in Hua Hin, and then terminate in Chumpon, where you transfer to their high speed catamarans for a one and a half hour trip to Koh Tao, then on to Koh Phangan about another hour later.
Seatran Discovery runs a ferry and daily excursion from Ban Rak in Samui, stopping at Koh Phangan and on to Koh Tao. They then move on to Koh Nang Yuan for a bit of snorkeling, move to Mango Bay for some more, plus lunch, then return to Koh Tao, Phangan and Samui. You can buy a ticket for the whole trip, or just use the boat for inter-island transfers. See their web site for more information.