The gateway to Thailand is Suvarnabhumi Airport. The IATA code is BKK. When you arrive at Bangkok Airport, look for the Airport Rail Link (ARL) signs. At platform 2 in the airport station you get on the train towards Phaya Thai. You drive a little more than 20 minutes and have five stops along the way.
At Makkasan Station, you need to transfer and switch to the MRT Blue Line. This is Bangkok's subway. The subway station is called Petchaburi and it is about 8 minutes walk. Look for MRT signs. In the subway station you have to get on platform 1 and take the train towards Lak Song. Don't worry: you can't really get lost, because the public rail transport is very general, because the trains go either in one direction or the other. If you got on the wrong way, just go back.
With the subway you travel another 20 minutes and have to wait 10 stops. Get off at the Sanam Chai stop. From there you continue by boat across the river. Near the metro station is a canal that flows into the river. It only takes about five minutes to walk down to the shore. There you are at the Rajinee boat stop. Numerous boats of different lines stop there. It is important that you drive upstream. So to the right.
The Rajinee River Bote Pier
You can board the boats that fly an orange flag or no flag at all. This marks the lines. The journey takes a while and you get to see many of Bangkok's highlights from the water as you drive by. Payment is made to the boat conductor on board. You tell him where you want to go. The ride only costs a few baht.
I present two destinations for arriving by subway and boat from Bangkok Airport here:
If you want to travel further to take the train west, for example to take a motorcycle tour through Thailand's green province of Kanchanaburi, you have to go to Thonburi Railway Station boat station. In Thai this means something like "satanii lot fai thonburi". Trains depart only from Thonburi station for Kanchanaburi.
Thonburi Railway Station River Boat Pier
Alternatively, you can also get off on the eastern side. Where the backpacker mile Khao San Road is. For this you have to continue by boat one stop and get off at Phra Athit station (pronounced "pra atit").