Getting from the airport to Bangkok city
Bangkok has 2 airports, Suvarnabhumi airport (Location here) on the north eastern side of the city, and Don Muang airport Location here on the north western side of the city. Suvarnabhumi is the bigger airport and the main international airport of the city. Don Muang is mainly used for budget airlines like AirAsia, NokAir, and ThaiLionAir.
Getting from Suvarnabhumi Airport to Bangkok
The 2 most common options are either taxi or train.
To get a taxi, go down to the taxi level at Floor 1, get a ticket from the taxi counter, and take it to the driver. Once you tell the driver your destination, they should start the meter and take you to your destination (meter starts at 35 baht). If they insist on a flat rate, refuse and if they won't start the meter get out and get a new taxi. In the past it has been common practice for taxi drivers to insist on flat rates to get extra money from passengers when the law states that the taxi driver must use the meter.
The total cost from the airport to your destination in town will be the meter cost + 50 baht airport fee. If there is a lot of traffic, the driver will also ask if you want to use the tollway, which will cost approximately 70 baht, which you pay to the driver then (the tolls booths will have change for any sized bill). A typical meter from the airport into town will be 200-300 baht, depending on traffic.
To get a train from Suvarnabhumi, go down to the lowest floor and follow the signs to the Airport Rail Link City. It leaves every 15 minutes, takes 25 minutes for the whole distance because it makes 6 stops in between, and costs 45 baht per person. It takes you to Phaya Thai BTS station, where the BTS can connect you to the entire central city. From Phaya Thai, you can either take the BTS to your destination if close to the BTS or MRT lines, or take a taxi from there to your final destination. If you take a taxi from there, again insist they use the meter (and if not just get a new taxi) - many taxi drivers will try to charge tourists flat rates to make extra money.
Getting from Don Muang to Bangkok
There are a few ways to reach Bangkok's city center from Don Muang, in which the bus and BTS combination is the best value and most time savvy.
To get a taxi- From outside the terminal you can bargain for a taxi to city center, or opt to use the meter. The fare is about 300 Baht.
To get the State Railway Train- From Don Muang you can take the train to Hualamphong, and then switch to the MRT. However, the trains are often late and this option can be time costly.
To get the bus and BTS- From outside the terminal catch the A1 bus, and switch to the BTS at Mochit station.