Rule #1 is to get out of their way and just wait for their expressions. It is more or less like nature photography, where you wait for hours to get the right shot (easier for you because babies are slower than birds :) ). You should be able to anticipate the emotions and have your subject in focus before the onset of the emotion..
I have shot a couple of birthday parties and it is fairly easy to do. Every kid in the world will smile and their eyes will light up when they see a candy/cake/balloon/sweets/gift etc.. If a kid notices you, he/she'dshe may be very curious about your gear... use it to your advantage.
Use follow focus and rear curtain flash sync. That will give a nice sense of motion to your pictures..