I started doing this 15 years ago when I first started to travel to Asia and have done it for every hotel stay since. The minute I walk into a room I drop my luggage in the washroom, remove the bedding and check for bedbugs.
Why the washroom? Bedbugs typically hate all tile hotel washrooms and there are very few places for them to hide in there. This make the washroom the safest place to store your luggage until you finish with your bed bug check.
How do I check for bedbugs? remove the bedding on the mattress and inspect the seams of the mattress (all around on both sides). You are looking for blood stains, small black dots (like pepper) or actual bedbugs. Look for the same signs under the mattress (between the mattress and the bed).
Next look immediately around the bed. Behind picture frames, mirrors, headboard, under telephones and alarm clocks. Check couches and chairs. Last but not least, check the hotel room's closet.
BedBug Registry (link) is an interesting website that allows people to report seeing bed bugs. Check it before you leave.
What should I do with my luggage and clothes? Never put your luggage on the mattress. If there are bedbugs, they may hitch a ride and infest your house. always store and open your bag on something unupholstered like a desk, nightstand or a purpose built luggage holder. Never leave it on the floor. Try to keep your clothes inside your luggage at all times.
What else? There have been cases of bed bugs being found in airlines, taxis and trains. Companies (Like BugZip LINK) make zippered bags to store your luggage to protect from these infections and from infection in the hotel room. Personally I have never used these but I know people who do. Another cheap solution is a simple garbage bag you can tie and untie.
Look for white powder around/under the bed, in closets, dressers or anywhere else in the room. This could be insecticide which may indicate the room was infected. Insecticide isn't good for your health and I wouldn't stay in a room that may have been infected (regardless of what the hotel staff says). If you see a white powder in the hotel, ask the hotel staff about and switch rooms. And when you move rooms, make sure it it at least 2 floors away from the "infected" one since bed bugs can move between walls and through electrical sockets.
If you find something or even suspect something, speak up. Let hotel staff know and ask for an immediate room change. Be as verbose as necessary to make sure they take care of you properly. Remember bed bugs can transmit germs from customer to customers even antibiotic resistant ones.
What can I do to avoid bedbugs in hotels? Before starting your trip, call up the hotels you will be staying in and ask about their bed bug prevention program. Most reputable chains have a program of constant inspection and a deal with a local pest control company.
What to do when I get home? When I get home, everything get's put into the garage and all clothes get washed in warm water and detergent (regardless of having worn something or not). Bed bugs die in the hot water. Anything you can't wash at home should be taken to the dry cleaner immediately. I then use my air compressor to blow air in every nook and cranny of my luggage (outside my garage). Then the last step is to vacuum the luggage before bringing it into my house for storage.