Hi- Our suggestion for buying a bike for a child would be to consider that they grow pretty quickly. Don't buy a bike that is too big, to try to save money for later, or as everybody's mentioned, a new, cheap department store bike that won't hold up. Consider (instead of buying a new bike at all!) seeking out an older used bike, especially from a community bike cooperative or a yard sale. Quality is often better and weight is less. Don't forget a helmet for your kid, whatever you think about helmets for yourself.
If you don't know how to maintain a bike always take it to your friendly local bike shop to tune it completely and fit the seat correctly for your child. Friends often also are outgrowing bikes that are not getting handed down to siblings- ask around and see who is moving up to a new bike.
Our three kids have never had a new bike and love the ones that they ride. Our 10 year old finds that the older upright style bikes we get at yard sales (26 in wheeled Raleigh Colt or LTD3 or 24 inch wheeled Schwinn Varsity) are easier to ride than the newer low bar bmx/mountain style bikes with knobby inefficient tires, at least for where he rides here in the city. There are few good bikes that a 10 year old can ride until high school, even if that's what you really want. You could look at folding bikes like Dahons or similar I guess if your child likes them.
New style kids bikes are often too long for the height, so kids are too stretched out toward the handlebars. They can't make turns or react to braking well then. Arms should always easily reach the handlebars with a relaxed and comfortable posture. Kids should always be able to step down on a bike with their feet flat on the ground, usually sliding off the seat to do so. Cross bars should clear a child's inseam by two or three inches standing over the bar.
Oddly, even on a bike that's too big, kids usually sit too low. If their knees aren't almost straight at the bottom of the pedal stroke, think about raising the seat or moving up a size. It makes pedaling much easier.
For a new rider, start with a balance bike or make your own out of a very lightweight kid’s bike by taking off the cranks. Don’t buy a real bike at this stage. They are all too heavy and most are identical, regardless of price. Forget training wheels - they just delay learning to ride.
Once your kid can pedal, something with a coaster brake is good. We like older style, more upright postures more than the current knobby wheel style. You start with a 12 1/2 in wheel. When the kid seems as big as the bike you have waited too long - move up to 16 in wheels, and add a hand brake.
Next step is 20 inch wheels and at this point you should have two hand brakes to encourage learning to ride, maybe with a coaster too, and maybe more than one gear. Our kids have several gears, lights, reflectors, a rack, two hand brakes and smoothish tires at this stage, none of which costs a lot. Remember that BMX bikes are an option here, but they don’t have many gears and the tires are impractical.
At 24 in and 26 in sizes your kid will tell you what he or she wants, not the other way around. Keep them in their helmet. Good luck.