I'm taking a photo each day of my daughter. She's now almost 18-months old and moves around like crazy, so I can definitely relate to your post. 
First off, be sure that you're using the AF point closest to the eyes of the child. It's going to be less sensitive than the center point, so you'll need to make sure you're not in a dark room.
Make sure it's in AI focus mode. Personally I use one-shot rather than burst, which helps me to work on predicting when to press the shutter. I find that you'll end up with a higher keeper rate if you're more aware of the child than if you're just blasting burst after burst. Put the AF-point on the eyes/face and half-press the shutter. Wait until you like the framing and take the shot.
Finally, the easiest way to get a higher keeper rate is to use f/5.6-f/8 (or an even smaller aperture). Of course you'll have a trade-off on shutter speed, but a flash will help a lot as well.
I use flash for 99% of my photos (bounced using the FlipIt) and use Manual whenever indoors. I use ISO200, 1/125 and f/4-8 depending on the background.
Hope that helps, the best thing you can do is practice a lot and try to notice when your child is just about to move, blink, etc. You'll see some patterns and you'll (hopefully) have an easier time getting just the right shot. 