This post is modified from something I posted on the Treehugger forums, only SEVEN WEEKS into the bike commuting adventure. I recently was re-directed to it and discovered that it (mostly) still holds true, and hope it helps someone out there who is just starting out...
*Helmets aren't required by law in Los Angeles if you're over 18... but it's up to you to determine your own comfort level. For a beginner, I would advise always wearing one, because while you're learning the rules of the road you're simply more likely to get in someone's way and end up in a collision. These days, I wear my helmet for rush hour only, because rush hour drivers are angry and my coworkers worry too much.
*Bike paths are great, but few and far between. While you're riding or driving, start taking note of which streets in your area have bike lanes, sharrows, wider than normal shoulders, or disallow parking during rush hours (which gives the street an extra lane). Once you've noticed which streets have these features, plan your routes to use them when possible.
*You probably shouldn't ride the same route you drive. What you look for in a car (a straight shot to your location, many lanes, high speed limits) are not the same things that you want in a road when you're a bicyclist - (wide shoulders, slow speed limits, even gridlock that you can ride past with ease). I tend to look at google maps in 'hybrid' mode to help plan my routes using smaller streets to get me to my destination. I know google just came out with a bike maps feature, but it usually puts in too many twists and turns for my comfort level. For a beginner it might be great though, since it definitely favors low-traffic roads.
*It is actually safer to ride on the street than you think. It's safer than riding on the sidewalk, because cars see you easily when you're on the street, while sidewalk riders crossing through intersections appear to come out of nowhere from a car's perspective and are more likely to be hit. Start with smaller streets, residential neighborhoods and such, and you'll eventually begin to feel safer. I promise you, when I started this, I would rather walk my bike through a crosswalk than make a left turn, got scared by every car that came near me, and was generally terrible. Now I prefer to make a left turn from the lane on most occasions (except where I don't have enough road to get left safely), know which streets I like to ride on and which to avoid, and am generally 1000% more comfortable being a part of traffic.
*Signal! Most drivers will look at you like you're a moron out there waiving your arms around, but even if they don't understand your signal (left arm straight out for left turn, right arm straight out OR left arm bent up like a wave for right turn), it will at very least draw their attention to you at the appropriate time.
*If you are at an intersection and you've pulled up alongside/in front of a car while waiting at a red light, turn your head and make eye contact with the driver. This forces them to adknowledge your presence, and they're less likely to push you off the road.
*The cars moving in your same direction are not the ones which are most dangerous to you. The most dangerous cars are those parked to your right, which may open a door without warning, without looking. This is caled "getting doored" or "winning the door prize", and is much more likely to happen than an accident with a moving vehicle. After the door prize, the next most dangerous thing is cars which are turning in and out of driveways, because they are paying attention to other cars but can sometimes fail to see you. I tend to slow down in areas where there are a lot of business driveways, so that I'm sure than I can avoid a car that either doesn't see me or underestimates my speed in a pinch.
*Don't run red lights. While it is illegal, many of us treat "Stop" signs as "Yield" signs, but always always always stop for a red light.
*Related to the above: you CAN get ticketed on your bike! Running red lights, not having proper lights on your bike at night, or drinking while biking, can all get you a ticket.
*Don't ride against traffic. Dumbest, absolutely the single dumbest thing you could do. When you do this, you're now at risk for a headon collision which is worse than the little tap you might get from behind the other way, and you put other cyclists in danger.
*get a good LED headlight and red taillight if you'll ever be caught out after dark.
*if you're on a street where there is parallel parking on the right, I know you'll get the urge to be as far right as possible, but don't weave in and out amongst the cars. For example, some people will ride in the parking zone if there are no cars parked there, and then they have to re-enter traffic as soon as they encounter a parked car. This causes you from the eyes of the people in their cars to frequently disappear and reappear, and freaks people out and risks you not being seen. The safest place to ride is as far right as you can get in the right-hand lane.
*likewise, in many cities you can choose to ride on the sidewalk or on the street. Whatever you choose (I obviously recommend street), stick with it. Riders who ride sometimes on the street and sometimes on the sidewalk are difficult to keep an eye on for motorists, and frankly anger them because of this.
*Consider multi-modal commuting. Take the bus to work and bike back to avoid arriving at work sweaty, or take the bus most of the way both ways and ride the last mile or two from the bus stop.
...most importantly, stay aware of your surroundings, and HAPPY RIDING!!!