The quickest way to Koh Samui is to fly — there are direct flights from Bangkok and Phuket, landing at Samui's own airport on the northeast of the island, historically a Bangkok Airways stronghold though other carriers now fly in via Surat Thani on the mainland.
Overland, the standard route is an overnight train or bus from Bangkok to Surat Thani, followed by a ferry — around two hours from the pier to the island. The bus takes roughly 11 hours; the train is slower but more comfortable if you can get a sleeper berth. Combined bus-and-ferry or train-and-ferry tickets covering the whole journey are sold at travel agents in Bangkok and at Surat Thani itself, and are generally the easiest way to book the trip.