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The most obvious way to get around Bangkok is by taxi---just be sure to ask the driver to turn his meter on before you get in. Say no to drivers who try to give you a lump fee.
The tuk-tuk or three-wheeled covered motorcycle has become a symbol of Thailand, and a ride in one is an absolute must in Bangkok. These vehicles don't have meters, so haggle for a price with the driver before you get in. If it seems to high, say no and go to the next one.
Bangkok is infamous for its traffic jams, especially during rush-hours. For short distances at times like these, motorcycle taxis are the fastest way to get around town. However, please be aware that accidents are not uncommon. If you decide to use one, do make sure to wear the helmet your driver gives you---it's the law. Again, haggle for a price before getting on.
For a safer way to beat traffic, Bangkok boasts an extensive elevated-rail and subway system, called the BTS and the MRT, respectively. These cover a wide swath of the city from JJ market in the north, far east to Bearing, and even west across the Chao Phraya River to Thonburi. Chances are, if you want to be there, there's probably a station nearby.
Finally, let's not forget that Bangkok was once the "Venice of the East." At Saphan Taksin you can connect to Bangkok's river taxi.