First, let’s break it up into a few stages.
Stage One — Getting To Koh Samui
To get to Koh Samui the easiest and most expensive way would be to fly from Bangkok. The only airlines that fly from Bangkok to Koh Samui are Bangkok Airways and Thai Airways. Bangkok Airways is usually the cheaper option while Thai Airways is the flagship airline of Thailand.
The way many get there is by taking a train or bus down to Surat Thani and then taking the ferry across. I’d recommend trying to get a bus that has it’s destination as Koh Samui from the new Southern Bus Terminal (if you’re at the Southern Bus Terminal right near a big mall and there is no air conditioning, that’s not it, that’s the old one). To get to the new Southern Bus Terminal in Bangkok the best advice I can give you is to Google the address Phutthamonthon Soi 1 in northern Thonburi which is on the other side of the river opposite Bangkok. Technically it is in Thonburi but it’s part of the extended Bangkok metro area. So, if you get a bus directly to Koh Samui you do not have to worry about how to get to the Koh Samui ferry and get across to the island.
If you do not get a direct bus ticket to Koh Samui try to get a bus-boat or train-bus-boat ticket from a travel agent directly to the island. You will wait for possibly long periods waiting to get a short bus to the ferry once you arrive in Surat Thani but at least you know everything is sorted. Try different agents to get the fastest ferry possible if speed is important to you. There are some private companies that do high-speed boats but this is not consistent so you need to search.
If you do have to worry about how to get to the Koh Samui ferry and get across from Surat Thani, take note of a few things. There will be taxis (probably tuk tuks or songthaew’s, the former you’ll see in Bangkok and the latter are pickups with benches in the back) at the bus station in Surat Thani which will take you to the ferry. The problem is, there are different ferries to take. Probably you will be taken to a travel agent who will book you on the ferry they send the tourists on, but if you can get a trip to Donsak through the songthaew or from a local travel agent, you will find the fastest ferries to Koh Samui there.
For more on getting there, the Koh Samui airport site has good information on all the ways to get to Koh Samui at http://www.samui.org/airport/
Stage Two — Staying in Koh Samui
You may want to find a nice Koh Samui bungalow to rest after the trip from Bangkok. I would suggest going to Chaweng Beach as it is the most populous area of Samui and the stunning beach calls out for your camera to take a picture. Chaweng is on the other side of the island from Nathon pier, the getting off point from Samui. There are many places to stay on that beach so I recommend just getting there and finding a place with a reasonable price that you like.
To get around Samui Island you will need to take songthaews or taxis. These drivers love to rip off Thai and foreign tourists alike. Since air conditioned taxis will charge minimum 100 baht to get anywhere, know you should never pay more than 60 baht for a songtaew. Bargain and stand your ground, as always, with a smile.
Stage Three — Getting to the Full Moon Party in Koh Phangan from Samui
From Samui, getting to Koh Phangan is easy. Simply get a ferry from Nathon pier. There are many travel agents across the road from the pier who will give you a fair price. Alternatively, if you are in Chaweng you can take a short trip to Maenam or Bophut on the north side of the island which have speedboats to Phangan.
Then, you can find your way from there. Have fun at the Full Moon Party on Koh Phangan and remember to be safe.