Bangkok is a vastly-spread city in terms of the things you can do there, having information about the transportation systems in Bangkok will give you a huge advantage.
For in-city transport, I would say that the busses are the best and cheapest option. They are really inexpensive, they are convenient, but they are usually crowded. A trip in a non-air-conditioned bus will cost you 10 baht, while in an air-conditioned bus you will pay 15 baht for a trip. The network is pretty dense and covers almost all parts of the city, so you won’t have any problem getting around in busses. Roaming around in taxis will cost you about 70-150 baht on each trip, the trip for airport to Khao Sa road is a bit more expensive, about 400 baht.
The SkyTrain and Metro will cost you 15-55 baht depending on your destination. A day pass will a better option if you are going to travel a lot, it costs 120 baht. There is also a Suvarnabhumi Airport Rail Link Express that will take you from the airport to the Khao San road in 15 minutes, costs 150 baht for a trip.