The Chao Phraya Express Boat is the cheapest and most scenic way of getting to Khao San Road. If you're coming from the city centre, take the Skytrain to Saphan Taksin station, where you can transfer onto the express boat at Sathorn pier. Then ride the express boat all the way north to Phra Arthit pier, which takes about 35 minutes and costs 13 to 18 baht. Make sure to get either the orange flag line or the blue flag tourist boat, as the yellow flag line skips Phra Arthit pier completely.
If you're staying in Khao San Road, the express boat is the easiest way to get to Rattanakosin and Silom. It's only a short (but confusing) walk from Khao San Road to Phra Arthit pier. First walk to the police station side of Khao San Road, then take a right and immediately a left into Soi Rambuttri. Walk along the bend of the street and take a right into Soi Chana Songkhram. At the end, take a right into Phra Athit Road, cross the road, and look for the small sign to the express boat pier (it is near Navalai River Resort). Bring a good map, have a good sense of direction, or ask help from a local if you get lost.
The most important piers are Tha Chang (for the Grand Palace), Tha Tien (for Wat Pho) and Rachawongse (for Yaowarat). All lines attend Sathorn pier, where you can transfer onto the Skytrain. From here, you can go to Silom, Siam Square, Sukhumvit and Phahonyothin. The Chao Phraya Express Boat stops running around 18:00 or 19:00 depending on the day, so you'll have to rely on other forms of transport in the evening.
The quickest (though not the most comfortable) way to get from Khao San Road to Siam Square, Sukhumvit and Ramkhamhaeng is by Saen Saep Express Boat. The closest pier to Khao San Road is Panfa Leelard, which is near the Golden Mount and Wat Ratchanadda. There you can get on the Golden Mount Line, which is a direct boat service from Panfa Leelard to Pratunam. At that pier you can switch onto the NIDA Line, that runs from Pratunam all the way northeast to Wat Sriboonreung in Ramkhamhaeng. A single trip from Panfa Leelard to Pratunam takes about 20 minutes and costs 11 baht. On the trip, you will pass the stops Talat Bobae (for the garment market of the same name), Sapan Charoenpol, Baan Krua Nua (for Jim Thompson's House), Sapan Hua Chang (for Siam Square) and Pratunam (for Pratunam and Ratchaprasong).