Digital cameras, especially smaller compact ones, aren't always the ideal thing to pick for shooting fast moving objects, especially those such as children. The Powershot however is a fine little thing, and you should be able to get some good photographs. Half pressing the shutter is something you should do to focus on an area - but if it's instant focus which you seek, that's a different league of cameras.
Rather then letting the camera set the settings for you, try shooting in Shutter or Apature mode, or full manual, assuming you know what to do.
Here's a small article from elsewhere to easily explain things, for starters, if you don't know yet
here