Don't try and bike fast. The pace of traffic is a leisurely slow and keeps you from sweating or getting in accidents with pedestrians, cars, holes in the road and other bikes. Do not assume someone is going to get out of your way - they may be assuming the same about you.
Renting a bike in Beijing might not be easy (although it is easy in tourist spots outside Beijing). A reasonable rent would be 10 to 20 kuai a day but you may have to pay more. You will likely have to leave a cash deposit as well. Ask at your hotel or corner shops. You can also try renting at places where bikes are sold. Someone may even offer to rent you their own bike.
A bike traffic light keeps things moving smoothly.
If renting proves difficult or you are staying for a while, you can buy a bike. Bikes cost between 150 kuai for inexpensive ones to 500 or more for the nice, big, heavy and indestructible Flying Pigeons models. I've owned two of them and they are the Mercedes Benz of the Chinese bike world. However inexpensive mountain bikes with lots of gears are gaining popularity. Gears are not that important though as Beijing is flat and easy to cycle. Also remember if your bike is too nice it could get stolen. Never washing your bike makes it look old and undesirable.
If your bike gets a flat tire or the pedal falls off, fear not as you are probably within a short distance of a bike repairman stationed at the side of the road. There is usually a sign and parts of bikes such as tire tubes hanging from it. Fixing a flat or mechanical problem should be inexpensive - under five kuai unless you're doing something major like getting a new seat or installing a deluxe bell.