By ship
If you're staying at Khao San Road, the Saen Saep Express Boat is your best bet to get to Siam Square. It is only a short walk to the Golden Mount in Rattanakosin, where you can get on the boat. The service operates every day from 05:30 to 20:30. The Golden Mount Line brings you to Sapan Hua Chang pier for Siam Square, and terminates at Pratunam pier for Ratchaprasong intersection. You can switch to the NIDA Line at Pratunam pier, which brings you to Chitlom, Wireless and on to Sukhumvit and Ramkhamhaeng. A single trip from Panfa Leelard pier at the Golden Mount to Sapan Hua Chang takes about 25 minutes and costs around 9 baht.
By Public Transit
Siam Square is the most central neighbourhood of Bangkok and can be reached by both the Sukhumvit Line and the Silom Line of the BTS Skytrain system. The easiest entry point is Siam station, right on top of Siam Square and in the middle of the Siam area. It acts as the exchange station between both lines, so expect a lot of commuters during rush hour. The Sukhumvit Line continues east and passes Chit Lom station, which is close to Ratchaprasong, and Phloen Chit which serves the road of the same name. One stop west of Siam on the Silom Line is National Stadium, a handy stop that's close to MBK Center, Jim Thompson's House and the cheap guest houses at Soi Kasemsan 1.
The Sukhumvit Line and the Silom Line provide direct connections with the districts of the same name. There is also a direct connection with Phahonyothin and the Chatuchak Weekend Market if you take the Sukhimvit Line to the north. If you're coming from Rattanakosin or Khao San Road, you'll first need to take the Chao Phraya Express Boat to Sathorn pier. From there, continue your way along the Silom Line to Siam station. The whole ride takes around 40 minutes but is easier than trying to find the right bus!
The quickest way to get to Siam from Suvarnabhumi Airport is by Airport Rail Link. Take the Express Line to Phaya Thai station, a ride that takes 17 minutes. Then switch onto the BTS Sukhumvit Line and ride it to Siam station. As the systems are not integrated, you will have to buy a new ticket at Phaya Thai station.
By bus
Siam Square is among the most congested districts of Bangkok. Combine that with the complexity of the bus system, and it's obvious that you're better off using other ways of transport. However, if you're coming from or going to the Grand Palace area in Rattanakosin, the bus is the only way to get a direct connection. Bus 15 starts at Phra Athit Road and then drives over Ratchadamnoen Klang Road for Khao San Road, Lan Luang Road, Chakkaphatdi Phong Road and Bamrung Muang Road before arriving at MBK Center, Siam Square and Ratchaprasong intersection. This bus then continues to Silom Road, so take it in the other direction if you're coming from there.
An alternative is bus 47, which starts at Rama IV Road in Khlong Toei and then heads to Silom and over Phaya Thai Road. Get off at MBK Center. You can also take this bus from Tha Chang in Rattanakosin near the Grand Palace from which it goes through Ratchadamnoen Klang Road for Khao San Road, Lan Luang Road, Chakkaphatdi Phong Road and Bamrung Muang Road. Get out at MBK Center as this bus won't continue its way further along Siam Square.
Bus AE3 from Suvarnabhumi Airport does not run anymore as of June 2011. If you still want to come by bus, you can take bus 552, which goes to BTS station On Nut in Sukhumvit. From there, you can continue your way with the Skytrain. It's a long ride though, and depending on traffic it could take between 1 to 2 hours.