If you want zippy shutter response then most point and shoot digital cameras are going to disappoint. My Canon G2 was the same. If anything was moving at all I never got the shot I wanted when I pressed the shutter. I got better at compensating but never quite nailed it.
Even a low-end digital SLR will beat most P&S on shutter lag. The Nikon D50, Canon Rebel and Rebel XT will all take the shot RIGHT NOW. You press, it takes. There may be others point and shoot alternatives that do better.
Try going to
DPReview and looking there. You may find a smaller camera that fits your needs too. I will say that I'll never look back now that I have my Canon Rebel XT. I get those fleeting moments and expressions on my kids face far more often and the flexibilty of the camera is just amazing. My G2 was nice but it failed on any moving subject.
ETA: If you don't do it already learn to "half-press" your shutter button. When you press just hard enough to partly depress the button but not enough to click through and fire the shutter you will cause the camera to do it's autofocus and exposure setting. Then you press just a bit more and fire the shutter. That will reduce lag greatly. That only works though on subjects that are not moving towards or away from you quickly. If they are then your focus will be wrong when the shutter fires anyway.