How to ride bike safely?

1) Check your equipment. This means:
Air -- are your tires inflated?
Brakes - do they work, are they clean?
Chain - is it clean, free of debris, and able to turn with no problems?
Check your brakes. Wheel your bike around a flat surface, pulling the brake levers on and off. If they do not work, contact your local bike shop and ask them what to do.
See if it is the right height for you. This is very easy as all you need to do is sit on your bike and make sure that the tips of your toes touch the ground (and nothing else).
Check your bike's tires. Put the whole tire in some water and see if there are any bubbles coming out. If there are any bubbles, it means your tire is punctured and you can get it fixed by a bike shop or do it yourself.
2) For roads that have traffic speeds under 35 MPH and/or a large shoulder (for example, many of mid- and up-state New York's 2-lane highways have shoulders that were designed for safe biking), ride on the road, in the direction of traffic. Share the road, however: If the road is wide enough that traffic can pass you, stay to the side and make it easier for the drivers to leave you room. Generally, it is illegal to ride your bike on a sidewalk unless it is a marked bike path. Additionally, roads are generally smoother, making the ride easier, quicker, and more comfortable.
3) Obey traffic laws. Although the driver hitting you is technically at fault, you are the one who will most likely be injured. Stopping for stop signs and watching for traffic greatly reduces the chances of a collision. Ride with the traffic signals; and if you are turning left, use the left turn lane. If you are not comfortable with this, walk or cautiously ride your bike through an intersection based on the crosswalk lights. Watch for cars turning right; drivers generally check only for cars when turning right, not pedestrians or cyclists.
4) Wear bright clothing, a reflective vest, or have flashing lights. These all increase your visibility. Motorcycles are required to have a front headlight on at all times, because their vehicles are smaller and harder to see. A bicycle is smaller yet. Night time riding usually requires a front white light.
5) Wearing a helmet is always a good idea when cycling. Consider wearing a bike helmet (local laws may require this) to augment your safety. See *http://www.cyclehelmets.org.
6) Use appropriate lighting at night. In the front, when it is dark out, turn on your headlight. During the day, a flashing front light is good because it attracts more attention, but at night, a constant front light is more appropriate, and it will provide adequate visibility to drivers. At night, a flashing front light is annoying. Also, put a red LED strobe or beacon on the back of your bike. It is OK for it to flash or have special patterns, because it disrupts night vision less than the white front lights, and because drivers do not count exclusively on your rear light to gauge their distance.
7) Use hand signals if you are going to turn. This is especially important for left turns, because they involve crossing traffic. Ignore what you were told in elementary school, instead, if you are turning left, extend your left arm; if you are turning right, extend your right arm. The signals that are all based on the left arm were devised for driving a car, a situation where only the left arm can reach out the window.
8) When you are passing pedestrians or other cyclists, it is essential that you warn them of your presence. Not only is this polite, but it reduces the chance of collisions, because they are less likely to suddenly swerve into your way, and they may move over, keeping you out of traffic. A bell, or a loud voice "On your [Left / Right]", "Passing" are good ways to attract attention.
9) Cycling at night can be extremely dangerous. Always travel on a well lit road or path with no debris or potholes. Go slower then you would in the daytime, as you still need enough time to react to danger and the lower visibility makes night cycling dangerous for this reason.
10) When riding past parallel-parked vehicles, leave enough room for the vehicle door to swing completely open just as you are about to pass. Getting "doored" can cause serious injury and/or death.