So, you will visit Bangkok ?
You will receive many offers for taxis and hotels just before leaving the arrival zone!  Ignore them. Say: "no thank you".  Most of these services are too expensive. After leaving the arrival zone without losing money, you may wish to first visit one of the currency exchange services.   That's ok!  Do it.
To go to Bangkok now, you will need a taxi. There are private taxis called taxi and public taxis called taxi-meter. Take a taxi-meter, because it is much more cheap. Some private taxis pay money to the airport, so they have a better location for waiting for passengers.
Money-Hint: Use the exit on the third floor (departure) and ask an incoming metered taxi for take you to your desired destination. These metered taxis generally come from Bangkok to carry passengers to the airport.  Here is the best (and cheapest) place to get transportation to Bangkok central.
How to "use" a taxi: First, tell the driver your destination.  If the driver likes that, tell him you want transportation "by meter".  If he still agrees, you have your first golden deal.  We highly recommend to enter the taxi now!  Let him use the "express road" and give him the tollway fee (40 Baht).  Finally, you will pay estimated 250 Baht (depending on destination).
If the traffic is very busy in Bangkok (it happens sometimes...), many drivers do not like to go "by meter". (This is illegal by Thai law and you can report them to any police officer.) Ignore them and move on to the next taxi which is willing to go by the meter. In the case you can't find a willing metered taxi you must bargain for the price. You should not accept more than 70% of his first offer.  Do not enter the taxi before fixing the price!   Finally, you will pay 300-500 Baht (depending on traffic and location).

The Airport has two terminals: International and Domestic.  You arrive at Bangkok Int. Airport. on the first floor (ground).   Departure (international) is on the third flour.  The second floor is for car parking and a "secret" 500m footway linking international and domestic terminals.
Be careful for booking unknown hotels at Bangkok Airport and also be careful of any offers (taxi, hotels, resorts) you receive after leaving the aircraft (you will get a lot of them).
Now you have two options to continue your travel:
You may go to the domestic airport for a domestic flight with Bangkok Airways, Thai Airways or Orient Thai Airways, or you can go to downtown Bangkok for a hotel, railway station etc...
Whatever you do: Do not forget your baggage!  This is very important if you want to go to the domestic airport.  Your baggage will not be redirected automatically at Bangkok Airport, so you have to pick it up, move to domestic airport and check-in your baggage again.  You would not be the first traveler who arrived at Koh Samui, while their baggage rests in Bangkok.
To go to the domestic terminal, you must leave the entrance hall, keep right and look for a sign: Shuttle-bus to domestic.   Here stops a bus for carrying travelers to the domestic airport and back - for free.
Alternatively you can walk on the passenger gateway at the second floor (about 10 min).   The domestic airport is quite small, so you will have no problem to find your check-in counter.