Before even booking a hotel or motel room, check for reports of bed bug infestation on the Bed Bug Registry, an online resource, or a site that reviews particular hotel/motel properties. Their information should be taken with a grain of salt, however, since the site merely collects reports but doesn't verify them, and of course not everyone who encounters bed bugs bothers to report it.
Before you do anything else in your room, inspect your bed or beds. Pay particular attention to the mattress and the headboard. The bugs themselves might be hard to spot---they are adept at hiding themselves---but if you see light brown bits of material, those are likely pieces that bed bugs have shed. Small black spots of bed bug excrement may also be seen in an infested bed. If you find such clues, insist on another room immediately, or go to another hotel.
If you wake up with small, itchy red spots on your body, take a flashlight and check---not so much for the bugs, who will already be hiding---but for small bloodstains on your sheets or mattress. Sometimes bed bugs can escape detection until after you are asleep. Fresh bloodstains are a dead giveaway for the bugs, however; get another room immediately.
If you have encountered bed bugs at all in a hotel or motel room, wash all of your clothes before returning home, even if you don't stay the whole night. Bed bugs are skillful at hitching rides in clothes and suitcases, and you could wind up with an infestation at your home or apartment---something you do not want.
Unless your suitcase is very expensive, dispose of it after an encounter with bed bugs. For expensive suitcases, a professional exterminator might be able to destroy hitchhiking bed bugs with pesticides.