Let’s start with the beginning, from suvarnabhumi international airport, as most of us arrive there first. My main concern when I arrive there is baggage, if I have 2 big suitcases I will go straight looking for a taxi. You will easily find the public taxi stand on ground level, near doors 3, 4, 7 & 8. Just queue, choose a normal or bigger car, get your ticket, follow the lane and meet your taxi driver. Inform the driver about your destination, show him a map or this post on your smartphone and be certain to double check he understood. 

There is a 50 bahts surcharge and you will have to pay for expressway costs. The process is regulated, taxis you get from here are controlled and must be certified by Airports of Thailand. Usually a drive to Khaosan road is about 300-400 bahts depending on traffic and expressway fees. 

If I travel light and am not too much tired of my flight I will go to get the bus S1 at ground level near gate 7. Service hours from 6am to 8pm. Fare is 60 bahts and the bus takes expressway so you will reach khaosan in about 1 hour. 

Another solution is to take the airport link, but you will have to stop to Phaya Thai BTS station then get a taxi from there, Khaosan road is about 5km from here but traffic is usually bad and taxis not really willing to stop or will try to take advantage on you. Not the easiest way for a first time arrival so I highly recommend the first 2 solutions above, especially if you have heavy suitcases and/or travel in group.