Shoot the Motion
Kids are in constant motion. Nearly by definition, shooting video of kids is shooting motion. While most still cameras and some video cameras have settings designed especially for shooting motion, with the iPhone you have to rely on your own skills.
One easy pitfall of shooting motion is to try too hard to follow the action. While in some situations that may be necessary, often it will lead to shaky and messy footage, which can range from "mildly difficult to watch" to "downright annoying." A good strategy for shooting kids in motion is to plant yourself (and the camera) firmly in one place, allowing the motion to move across the frame, as shown in Video 1. Even if the subject moves in and out of view, the stable image will be easy to watch. A good example of a time for this technique would be when a child is learning to ride a bike: Each wobble isn't worth catching, but a shot of a bicycle winding in and out of frame is fun.
By allowing the subject to go out of frame without following her, I wind up with a steady clip capturing some of her action, instead of a jerky one capturing all of it, which can be a very good trade for fun-to-watch video.
If following a moving subject is necessary (and it often is, as a subject who leaves the frame might not come back!), don't feel obligated to anticipate your subject's movement. In fact, resist the urge. Anticipating movement is another tempting and common pitfall, because it would seem that moving the camera to where your subject might go would allow you to catch her "landing," so to speak. However, by anticipating this obviously unchoreographed movement, more often than not end you'll end up with a poorly framed shot, or you'll need additional camera movement to correct. If you slowly track your subject instead, you can avoid those jerky movements at the end.