Since Samui is an island, your choices for getting there are somewhat limited:
Samui's award-winning airport was built and is serviced mainly by Bangkok Airways flying from Bangkok, other parts of Thailand, and Singapore. There are flights between Samui and Bangkok more or less hourly from early morning until about 10:00 in the evening.
For advice on getting from the airport to your hotel, see the Getting Around page.
A budget alternative to flying Bangkok Airlines is to fly one of the low cost carriers (see Getting Around Thailand for a current list) into Surat Thani or Nakorn Si Thammarat and then take a bus / ferry combination to Samui. The main drawback to this is that it will eat up more or less an entire day to get there.
There are two main ferry companies operating boats between the mainland and Samui. Most passenger ferries leave the mainland from the pier a short distance from Surat Thani town, while car ferries use the port at Don Sak. Passenger ferries arrive at the main pier in Nathon Town, while car ferries arrive at a special-built pier further south down the west coast (see map).
Ferries to Phangan can be caught at either the pier in Nathon Town or at Bophut on the north coast. Lastly, it's worth noting that there is a combination bus / ferry service from Bangkok. Buses take you from Bangkok to Chumpon and from there a ferry travels to Tao, then Phangan and finally Samui. One tour operator to check out is Lomprayah.