Riding With Kids
Since most of us learn to ride a bike as a kid, I think it's especially important that they learn to do it right the first time. This is a developing section, but I'll start off by noting the most important thing kids should do on a bike: WEAR A HELMET!
I know, none of us adults grew up wearing one, and we were fine. Aren't we lucky! Not all kids are, and head in juries are the most common outcome of cycling accidents where kids are concerned. When I teach kids' cycling classes, the first thing we do is fit their helmets. When a kid suggests they look 'dumb,' we remind them they'd look a lot dumber in a hospital bed with a tube up their nose!
That said, I also have to emphasize that they should not only wear a helmet, they should wear it PROPERLY. A properly fitted helmet sits level on the head, with about two fingers' width between the eyebrows and the bottom of the helmet. Straps should fit just under the ears, and you should be able to get no more than two fingers between the chin and strap. The Bicycle Helmet Safety Institute has a more detailed fitting checklist here.
As with anything, you can't prevent every accident or injury - but you CAN improve your odds.