1. How to sit . Before you start trying to move you need to know how to sit. You should not be sitting on your ass like a ton of bricks. You are doing something physical - engage your muscles! The saddle (sic. "seat") of a bicycle should carry maybe 20% of your weight; your hands should get 20% and your legs should get 60%. I think the best way to feel how you should be sitting on a bike is actually to get off the bike and get into the position that a sprinter gets in at the start of the race : squat down with your legs apart, put your two hands in front of you, now extend your legs and arch your back - that's a biking position.
Many people incorrectly think that they need to sit upright for their "back". Nonsense. Upright is actually very hard on your back, and if you sit with your weight on the seat it means you are transmitting every bump in the road straight into your spine. It's actually much more natural for your spine to be arched. Imagine yourself like a cat, your spine is bent, it has springy energy, when a bump in the road shocks you it just flexes the bow of your body a bit, no big deal (a proper biking position is kind of like the cat back position in yoga). (upright bicycle sitting is good if you have a bad *neck* - the forward leaning bicycle position while actually good for the back is bad for the neck when you look upward with it).
Never ever lock your elbows or your knees and just rest with them locked. They should be slightly bent at all times. Your muscles should always be engaged so that you are a big arch of muscles - never resting on bones or joints. You should be light on your feet with tension in your legs and your weight on your feet, not your seat - you should be able to "post" out of your saddle like an English style horse rider at any time. The way you make your ride soft is not by buying a big padded seat (which is very bad for you) it's by posting out of your saddle and turning your body into a bow.
Obviously you can't sit correctly if your bike doesn't fit you. This is too big of a topic for to cover in depth here, but I'll say a few things. Many road bikes sold to the casual market have way too big of a drop from the saddle to the handlebars. For casual riders you actually want the bars and the saddle pretty close to equal height. Sadly most bike shops either sell ridiculous casual bikes that are way too upright or racing bikes that have way too much drop, it's hard to find a comfortable efficient road bike but it's getting a bit better these days.
(aside) : Another issue with fit that I'll mention because most shops fail on this so hard : presumably you are a beginner or a semi-amateur like myself that goes in an out of shape; if you are buying a bike you are probably not in great biking shape at the time that you buy it, but you will be riding yourself into shape over the course of a season. As you get into better shape, you will want a different position on the bike, a more aggressive drop, perhaps a more powerful saddle position. What this means is you need some room to adjust in your bike sizing. You should never buy a bike that just barely fits you, or where you have to put the seat all the way down to the frame to make it low enough, or all the way up to the end of the seatpost to make it tall enough. It's important that you get a frame that is comfortable now in a more relaxed position, but has room to adjust towards more aggressive and still fit you. In general if you're deciding between a frame that's a bit too big (so there's very little exposed seatpost) or too small (lots of exposed seatpost) it's better to go with the frame that's too small, because you do have some ability to make it "act bigger" through the stem and seatpost, but you can never get smaller than the frame limits.
The other big issue is how your saddle contacts your body. You should feel the contact on your sit bones. You should not be sitting on the nose of the saddle with it jammed into your taint. The nose of the saddle is not for sitting on, it's for guiding you on and off the saddle when you post in and out of the saddle. When you sit back onto the saddle you should feel your sit bones contact the two padded wings near the back of the saddle (but not so far back that your thighs run into the saddle when you pedal). Your pelvic bone should be pretty close to vertical - not tilted way forward so that you are jamming your gonads into the saddle. Your gonads should basically not be contacting the saddle at all; you'll hear beginners say "ow it hurts my balls to sit on those hard skinny race saddles" ; well dumbass you're not supposed to sit *on* your balls. When you lean forward to reach the handlebars it should come from arching your back, not rotating your hips - your pelvis should stay vertical. Note that saddles are not one size fits all ; you should get a saddle that is the correct width for your pelvis and sit bones (women in particular should usually have non-standard saddles, but so should men with unusually wide or narrow hips).
You need to get comfortable in "ready position". When coasting, get your pedals level with each other, stand up on them so your butt is just barely off the saddle. Hold your bars with both hands, elbows and knees should be bent, back should be arched. This is the position where you have the most control of the bike and are ready for anything. This is how you should go over bumps, this is how you should be when you might need to brake hard, when you're around dangerous traffic, etc. Any time you are riding you should be able to post up into ready position at any moment.
2. How to brake . This is perhaps the biggest mistake that I see beginners make and maybe the most important because good braking is critical to surviving.
You stop yourself with the front brake. To stop quickly, you need to pull it *hard*. You need to get comfortable with this before you go riding at all, because when that car or pedestrian jumps out in front of you, you should be able to stop from high speed fast. Lots of beginners go off riding fast and can't stop themselves and it's a huge mistake. You need to hold the handlebar hard while you brake, your hand should wrap around the brake and the handlebar and you squeeze them together.
Beginners are afraid of pulling the front brake hard because they are not in good control of their bike. First of all, learn to sit right and be comfortable in "ready position". Now, when you brake really hard, hold the handlebars firm with both hands so they can't twist. The bike is going to stop suddenly which is going to fling your body weight forward and try to throw you off over the front of the bike. If you are in ready position and using your muscles that's not really a problem, it's only bad if your body is limp or your limbs are locked and you're just sitting on the saddle. When you jam on your front brake, you should push forward with both your arms, sort of like a push-up movement, but to push your body weight backwards towards the back of the bike; this braces against the momentum that will try to send your body over the front; you also want to crouch down on your legs to lower your body weight.
You should practice stopping with only the front brake to make sure you are comfortable with it. In the dry you are not going to skid on your front tire ever - you will skid on the back very easily which is why you don't use it for hard stops. You should be able to stop from 20 mph in about 10 feet. If you can't do that, you need to practice jamming on your front brake harder. To brake really hard when going fast, you want to get your hands down in the front vertical part of the drops, get your weight up on your legs and crouch down and move your bodyweight back to get ready. Don't go fast unless you can brake properly when going fast.
3. How to pedal . Now that you know how to sit and brake you can start thinking about moving. The first issue is how to pedal.
First make sure your leg is moving through the proper range of motion. Your seat should be at the right height so that your leg can extend almost all the way in the down-stroke, but not so that your knee straightens completely or you have to reach with your tip toes or sway your hips. Your hips should stay level, your knee should stay bent at all times, and your foot should stay roughly flat the whole time (a tiny bit of "walking" motion is okay). Again one issue here is getting the bike sized right for you; to some extent you can adjust this by raising or lowering the seat if that works with the rest of the bike geometry; however one issue that is often not addressed is the length of pedal crank arms; people with shorter legs should have shorter cranks so that their knees are not coming up too high at the top of the stroke.
The next big issue is cadence (how often the pedals go around). You want to be pedaling at least 60 rpm generally. Amateurs often pedal very slow cadence in heavy gears and have to stand up in get their weight into it to turn the pedals over. You want your pedalling motion to be a circle, not pounding like pistons that only push in the downstroke. You want to feel like your feet are just constantly whooshing around in fast circles.
4. How to ride with cars . If you're riding in the city you need to know how to be a safe rider around cars. Your main concern should be your own safety.
The most important thing is being predictable and visible. There are a few things that amateurs do really wrong.
First of all, you should ride in straight lines as much as possible. This makes it clear the line you are taking. In particular, do not weave in and out of parked cars. A lot of amateurs are scared of traffic and want to stay as far to the right of the road as possible, so they will weave in to the right when there's a gap at a driveway or whatever. This is a very bad thing to do. Pick a consistent straight line and stick to it.
Second, claim your space. Make yourself visible and take the road when you need to. Don't worry about blocking cars if you have to. For example if you see up ahead that there's a branch in the bike lane or something, don't try to just barely skate around it and leave the lane for cars. Instead pick a safe time and pull out all the way into the middle of the car lane and claim the space visibly until you pass the branch. Don't wait until the last minute before an obstacle and swerve to avoid it - claim the space early and make it clear what line you will be taking.
Don't be afraid of the cars that are moving along beside you. They are actually the least of your worries (but do worry about them if they are braking or moving erratically as it indicated they are trying to parallel park through you or turn right across you). Do be afraid of parked cars - don't ride too close to parked cars or driveways. Generally being out in the middle of the road where you are visible and can see other moving cars is the safest area to be. Being near obstacles and things that block vision like buildings or parked cars is the worst place to be.
Remember to watch the road conditions and be vigilant and don't be afraid to defend yourself and inconvenience cars. Most of my worst crashes have been due to junk in the road that I hit because I was watching the cars, or I was trying to be polite to the cars and was riding in the shoulder or the gutter or something.
5. How to shift . I won't say much on this (see Sheldon on Shifting ) but two things that I see almost all beginners mess up :
First, bikes are designed for you to use the front & rear derailleur in tandem, not just one or the other. You should think of the chain as wanting to always be parallel to the bike. You want to shift such that the chain moves directly sideways either inwards towards the bike (lower, easier gears) or outwards (higher, harder gears). The chain does not want to be at an angle to the bike. So, you should use the small front chainring for the lower (larger) gears on the back, then as you shift up through the sprockets on the cassette (in the rear) at some point near the middle you should then pop up to the larger harder front chainring, then finish shifting up through the sprockets on the back until the chain is all the way to the outside.
Second, your front derailleur almost certainly has a "feathering" adjustment capability of some kind. What this means is that the movement of the front derailleur is not digital - it's not just on the small chainring or the big chainring in a fixed binary state. You can "feather" the front derailleur to move it slightly without shifting to the other state. This lets you move the derailleur cage so that it doesn't hit the chain. Any time you are "cross gearing" at all (small front chainring to small real sprocket or big-to-big) (cross gearing is bad, it's the angled chain that I said to avoid in the first section) you can use the feathering to keep the front derailleur cage from rubbing on the chain.
A properly adjusted bike should be almost silent. If your gears make a lot of noise as you shift, either you are doing something wrong or it is out of adjustment.