From Suvarnabhumi Airport, take the Airport Link to go to Makkasan station and exchange to MRT subway to go to Hua Lamphong station, then take a taxi from here or take a bus, air-con bus number 40. It's start station that pass through Wongwian Yai and Charan Sanit Wong road. Expenses around 20-25 baht. We not recommend this route because it is take many time but if you have many time and want to get the new experience, it's ok for you.
From Suvarnabhumi Airport, take the air-con bus number 556. They service around 06.00 AM - 09.00 PM, expenses around 24-35 Baht.
If you stay in Sukhumvit area, you can take the air-con bus number 511 (orange color) to go to Southern Bus Terminal. Expenses around 20 baht and inform bus staff to drop off at this station (last station of bus route). Take a time around 45 minutes or more. (Depends on traffic.)
If you stay in Khao San road area, taxi is the best way because it's fastest.