No. 554 Suvarnabhumi - Don Muang: 
Suvarnabhumi - Ram Indra Road - Lak Si - Vibhavadi Rangsit Road - Don Muang.

No. 555 Suvarnabhumi - Rangsit (Expressway): 
Department of Employment - PTT Don - Muang - Vibhavadi Rangsit Junction – Rangsit.

Shuttle Buses
A free shuttle bus will take you from the Domestic Terminal to the International Terminal within Don Muang.

Airport Limousines
Limousines are operated by three companies, Airport Associate Limousine, Thai Limousine, and Phrapirab Limousine. They all have the same fares, starting from 650 baht for downtown Bangkok. Contact them at the International Arrival Halls and Domestic Passenger Terminals.

Taxis
You will find an airport taxi stand just outside the arrivals building where you can book a taxi-meter. Most destinations in downtown Bangkok cost around 200-250 baht. Besides the meter fare, there is also an airport surcharge of 50 baht and a toll of 30/40 baht if you take the highway toll.

Airport Buses
There are four bus routes which cover four main areas of Bangkok:

Route A1 (South): Don Muang – Silom
Route A2 (West): Don Muang – Sanam Luang (Grand Palace)
Route A3 (East): Don Muang- Phra Khanong (Sukhumvit)
Route A4: Don Muang – Bangkok Railway Station (Hua Lamphong)

Air-con & Regular public buses
These buses are the cheapest method to get from the airport to downtown Bangkok, much cheaper than the airport buses, however they can be pretty crowded especially during rush hours and not practical, especially if you have a lot of luggage.

There are five routes for air con buses going from the airport to different main areas in Bangkok.
 
Route 4 Airport – Silom Rd.
Route 29 Airport – Victory Monument – Bangkok Railway Station (Hua Lamphong)
Route 10 Airport – Victory Monument – Southern Region Bus Terminal
Route 13 Airport – Sukhumvit Rd. – Eastern Region Bus Terminal

Train
You are also able to get into Bangkok from the airport by train. The Don Muang Station is located across the highway and opposite the Amari Airport Hotel. The trains arrive/depart from the airport station every 30/60 minutes, but this can be longer at weekends. It can take an hour or more during rush hour to Hua Lamphong, the main station in central Bangkok.

Click here for what to see and do in Bangkok.

Flight schedules from Bangkok:
Bangkok to Buriram
Bangkok to Chiang Mai
Bangkok to Chiang Rai
Bangkok to Hat Yai
Bangkok to Hua Hin
Bangkok to Khon Kaen
Bangkok to Koh Samui
Bangkok to Krabi
Bangkok to Lampang
Bangkok to Loei
Bangkok to Nakhon Phanom
Bangkok to Nakhon Si Thammarat
Bangkok to Nan
Bangkok to Narathiwat
Bangkok to Phuket
Bangkok to Roi Et
Bangkok to Sakon Nakhon
Bangkok to Sukhothai
Bangkok to Surrat Thani
Bangkok to Trat
Bangkok to Ubon Ratchathani
Bangkok to Udon Thani

Flight schedules to Bangkok:
Buriram to Bangkok
Chiang Mai to Bangkok
Chiang Rai to Bangkok
Hat Yai to Bangkok
Hua Hin to Bangkok
Khon Kaen to Bangkok
Koh Samui to Bangkok
Krabi to Bangkok
Lampang to Bangkok 
Loei to Bangkok
Nakhon Phanom
Nakhon Si Thammarat
Nan to Bangkok
Narathiwat to Bangkok
Phuket to Bangkok
Roi Et to Bangkok
Sakon Nakhon to Bangkok
Sukhothai to Bangkok
Surat Thani to Bangkok
Trat to Bangkok
Ubon Ratchathani to Bangkok
Udon Thani to Bangkok