Sorry this is so long, but....
This is Karen's no-crash method of teaching kids to ride a bike. Feel free to share with anyone who might be interested. I've taught several dozen kids to ride a bike this way. Not one of them crashed during the learning process. I can and do absolutely promise them that they will not crash while they are learning. I also warn each of them that they will crash 2-3 weeks later when they get overconfident. This first major wipeout happens like clockwork. However, by that time they're confident enough on the bike that it doesn't discourage them from biking. Learning time from step 5 to independent biking has ranged from 5 minutes to a few days, depending on the kid.
Step 1: Teach them how to use pedals. Make sure your child has mastered the art of pedaling. Most kids learn this on a trike when they are small. If your child can't propel a tricycle, they aren't ready to learn to ride a bike. This is the only situation in which I recommend training wheels: if a child is too big for a trike but can't yet use pedals. Otherwise, avoid training wheels like the plague.
Step 2: Adjust the seat so that the child's feet are flat on the ground when they are sitting on the seat.
Step 3: Remove pedals and training wheels (if any). Be forewarned that if your child is accustomed to training wheels, they may hop around and complain a lot during this stage. Ignore them. If you can't get the pedals off, a bike shop should be able to do it for you cheap or free. Put the pedals in a safe place so you don't lose them!
Step 4: Let your child use the bike as a balance bike. Ideally, have them practice on dry concrete with a few small hills. Grass may be softer, but the traction is terrible and wheels are more likely to slip. Bike wheels grip concrete best. A few small hills are nice for practice, but remember that balance bikes don't have brakes, so big hills are a bad idea! Let them play on the bike until you notice them doing long swoops and glides. For some kids this is an hour. For other kids this is six months or more. Don't push them. Just let them learn to move comfortably with the bike on their own schedule.
Step 5: Cement the lesson of moving comfortably with the bike. Holding onto their handlebars, ride them over rough terrain and in tight circles. My dad rode me up and down inside irrigation ditches. I ride them over over bumpy ground and in tight circles. Bikes ridden properly lean when they turn. A good biker leans with the bike. A panicky biker leans against the bike and crashes. Teach your child to trust the bike, to move with the bike.
Step 6: When everyone's ready, reinstall the pedals. This is the first big “lesson” day, and probably the only “lesson” day. You need dry level concrete with a lot of space in all directions. An empty parking lot is ideal. A dead-end street is okay. A sidewalk is not good: too narrow. Most driveways are too short and too narrow. Avoid grass, it's too slippery.
Step 7: Teach the child to ride in a straight line. Stabilize their body and help them get started. Hold the child, not the bike. If the bike crashes, whisk them up and out of the way. I promise my students that the bike might hit the ground, but they won't. Remind them to keep their head up and looking at a distant target. Remind them that they need to go reasonably fast if they expect to keep balance. Run alongside them holding their body until they've got the hang of it and are trying to ride faster than you can run. Go from a strong grip on their body to a light grip on their body to a grip on their shirt. Remember to let them move with the bike and not try to keep them rock-steady. Don't correct every wiggle. Just make sure they don't hit the ground.
Step 8: Teach them how to stop. Instruct them in using the brakes. Remind them that they need to get a foot down immediately or they will fall over. When they don't get a foot down fast enough, let them fall partway so they get the muscle memory of falling....just don't let them hit the ground. Also remind them that a skid means they have lost control. Their goal is to come to a smooth stop with no skid and have a foot down before the bike tips over.
Step 9: Teach them how to start. Figure out whether they are right-footed or left-footed. Usually people lead with the same foot as they use to kick a ball, but have them try both ways and see which feels better. Set up the bike so the pedal is a bit past vertical on the dominant foot side. Remind them to stand on their non-dominant foot, push hard with their dominant foot, and get their other foot on the pedal immediately. Keep a firm grip on the child's body, because their steering will be terrible. After they've got the hang of the sequence, tell them to stop watching their feet and look at their long-distance target. This will improve their steering dramatically.
Step 10: Put it all together. Have them start, bike in a straight line , then stop. Make the runs long enough that they feel like they're riding and short enough you can keep up.
Step 11: Practice turns. Have them start up, go straight for a while, then turn left. How about turn right? Call out directions to them like “steer towards the big tree.....now lets head back towards the large rock over there.” Remember, you're still holding onto at least their shirt through this whole process. You're not actually putting any force on their body, but they can feel that they're “on a line” and you are ready to catch them if they fall.
Step 12: Transition to solo riding. At this point your child can reliably start, ride, stop, and turn without you providing any support. They're probably also getting really irritated that they can't ride faster than you can jog. If they feel ready, let them solo. The first solo is usually a supported start to a solo straight ride and solo stop. Next, move to a solo start, straight ride, and solo stop. When they're confident at that, have them do a solo start, straight ride, wide turn, ride straight back to you and stop.
Step 13: Talk about traction. Warn your child that wheels grip differently on dry pavement, wet pavement, dry grass, wet grass, mud, gravel, and sand. Remind them to be extra-careful on any surface other than dry pavement. Be especially cautious around gravel or sand, which are notorious for making bike wheels slip unexpectedly.
Congratulations, you've got a bike rider!
Put in a good supply of bandages and antibiotic cream and make sure you insist on helmet use. The first wipeout is usually a biggie. I've never had a kid break a bone in that first wipeout, but they usually lose a lot of skin.
Topics for follow-up lessons: Make a obstacle course and have them steer around cones and stop on a line. Teach them to ride with the right hand only. Then teach them to use hand signals for turning. Teach them the rules of the road. Bikes have most of the same rules as cars, but a lot more vulnerability. Teach your child to assume all cars are driven by poor drivers who don't see the kid on a bike unless they have made clear eye contact with the driver. If your child will be riding in low-light conditions, make sure they have appropriate lights for visibility.