Fastest and most expensive.
-Fly direct from either BKK airport(1 hour) or Singapore airport(2 hours). Typically about 5000 baht(165usd) each way. The airport on Koh Samui is about 20min from the course.
-In Bangkok fly from either BKK or DMK airport to Surat Thani (URT). After arriving take a 1:40 bus/van to the ferry, and the ferry takes about an hour. I recommend booking with AirAsia which gives you 1 price for flight + bus + ferry and while it takes 6-8 hours it is fairly easy. Search DMK to koh samui to book this. Which ferry pier you arrive at will depend on which ferry you take, but all with have songtao(truck taxi) waiting when you arrive. Flight without bus and ferry included will be 700-1000 baht($22-32 usd) and with bus and ferry about 1300-1500baht each way if you book ahead of time. Prices will go way up for these near the date so book early
-From Singapore or Malaysia you can fly from KUL airport direct to Surat Thani, and from there take bus or van to the ferry. For this route even though it is AirAsia flight the combo ticket is not available. This flight can be had for under 1000 baht each way.
-Bus. From Bangkok you can take a 12-14 hour bus direct to Koh Samui island. Leaves in the evening and arrives on Samui in the morning. Cost is about 900 baht. I recommend using 12go.asia to look into these. Also any travel agent in a tourist area of Bangkok will be able to help you.
-Train. There are trains from Hua Luampong station to Surat Thani that take approximately 11 hours. Non AC seats can be had for as little as 300 baht, and a sleeper bunk is 840 including a meal. However If you are not familiar with Thailand I don’t recommend this option as you may spend whatever money you might have saved to get from the Train station to the ferry.