First, make sure the background is uncluttered.
You want the focus to be on your child, and not the bookshelf or people behind him.
When the background is too "busy", it can distract the eye from your subject.
Move him to another spot, such as a bare wall or if you're outside, in front of some bushes or a wall.
Next, get closer.
One of the biggest mistakes parents make is to think they have to include their child's entire body and everything around them.
Either zoom in with your lens or actually walk closer to your child, to focus in on his face.
In this way, you'll capture precious details of your child's face at this stage.
Finally, the most important thing to do to improve your photos is to turn off your flash! The flash on your camera is your worst enemy when it comes to taking nice snapshots of your child because it flatten out your child's face and cast a horrible shadow behind them.
So turn off your flash and photograph your child with natural light.
Put them near a window, or outside in the shade.
You'll be amazed at the difference in the quality of your photos! Cluttered backgrounds, the subject being too far away, and a direct flash are tell-tale signs of an amateur photograph.
Practice with the tips I outlined above and in no time at all you'll see dramatic improvements in photos of your children.
Remember, though, even if the all the conditions are not perfect every time, be sure to get the shot anyway.
What's most important is that you preserve the moment forever.