Forgot to mention perhaps the most important rule of all.  A video camera is for ACTION!!  Make sure that you keep it that way, unless of course you are getting comments on those WATCHING the action going on before them. 

Keeping the camera rock solid without a tripod.  It's quite easy once you get used to it.  One hand underneath the lens and one hand on top of the camera.  Feet comfortably astride.  You will soon find your correct and comfortable stance.  No sudden movements with camera or your own body.  Follow the action as smoothly as possible.  This can sometimes be difficult with the fast movements of kids, but you'll soon find yourself able to anticipate.  Above all those considerations, and one which I should have put first, is to make sure your lens is sparkling clean and polished with a proper lens cloth which you can get from a camera shop.  Good Luck