As you may or may not know I taught myself photography by shooting my own kids.
If you really want to capture your kids true spirit the most important rule is not a big camera or complicated lights but anticipation.
The photographs above were taken with a tiny Canon point-and-shoot. It was so bright at the beach I couldn't even see the screen on the back of the camera but i knew that if I got them both in the frame that I just had to wait for the right moment.
For example, the top photo is Isabel taking the sand sifter from Claudia with a let-me-show-how-to-do-this bravado. The photo is all about the look on Claudia's face because she knows she wont see the sifter again for a while, or until we intervene. It is such a classic moment between brothers and sisters. (to really get the shot, click on the photo to enlarge the image)
In the bottom photo, the snap on Claudia's top suddenly broke and she has this surprised/embarrassed laugh that shows just how cool she is and able to laugh-off her own embarrassment. Those are moments you can't script or re-shoot but when you see the elements coming together (like the two of them playing nicely with each other) then you just have to place yourself in the position to get the right light and angle. The photo "That Moment" posted July 14th was taken with the same idea - I saw the perfect light and the activity of people running into the show so I stayed in one spot and just kinda waited for something to happen. Most of the time nothing happens, but the few times it does the reward can be really great.