Bangkok has thousands of cybercafes all over the city. In the tourist areas around Silom, Sukhumvit and Khaosan Road are numerous enough that you need not map them out ahead of time, you can walk down the street and find one in minutes--if not seconds. A walk outside might not even be necessary, many tourist accommodations offer email services on premise and some provide a few minutes of access a day for free to guest.
For those of you travelling with a laptop, wireless hotspots are widely available, some for free, others for a fee. Many upscale hotels offer in-room wireless services (again, usually for a fee).
Outside of the main tourist areas there are many Internet cafes which cater to the locals; to locate one, look for a place where there is heavy foot traffic and you will, most likely, find one.
Internet cafes in the tourist areas generally charge 0.5 to 3 baht per minute. If you are planning to do a quick email check, beware--many have a "minimum charge" that is equivalent to about 10 minutes of use.
Most of the larger Internet cafes have upgraded to high-speed DSL lines but that does not mean that your connection will be "peppy." Often international traffic going over long distances has limited bandwidth upstream which means that the Internet can come to a crawl during peak hours.
Many Internet cafes offer other computer-related services such as printing and CD/DVD burning for a nominal fee.