Your problem won't be shutter speed, but rather shutter lag. Research that.
Ideally for capturing photos of kids or anything else moving you'd need a DSLR, because those focus much faster and have minimum shutter lag. Typically, a compact camera has slow focus, and even when it's prefocused, some cameras have a slight delay between pressing the shutter button and when the actual image is captured. DPreview has some good info on that.
These two are the responses to listen to. A "mere" 1/1000 second exposure time is fast enough to capture racing cars at full speed on a track. In most cases, 1/500 second is perfectly acceptable for "freezing" human motion, hell, in most cases 1/250 second is adequate.
The issues here are:
1) Most point and shoot cameras won't push much past ISO 400 equivalent speed (the higher-end point and shoot cameras will, but still not as cleanly as DSLRs).
2) Indoors, you will be hard pressed to see shutter speeds above 1/100 second from most point and shoot cameras because there's a lot less light indoors than outdoors (this isn't immediately obvious since our own eyes do a pretty fantastic job of adjusting to ambient light).
3) With less light available, focusing is more difficult, even for the inherently better autofocus on DSLRs, so the lag time becomes longer when indoors.
4) Point and shoot cameras, even when pre-focused, have more of a lag time between when the shutter is pressed and when the picture is actually taken, which is sometimes mistaken for a slow shutter speed.
5) The more you zoom in, the "slower" the lens will get. Lenses, in most cases, have a smaller maximum aperture when zoomed in, so using the zoom further slows down the shutter speed because the lens is letting in less light.
6) Anti-shake only helps with camera shake, which happens at low shutter speeds. If you're in a situation where anti-shake will be helpful, the shutter speed will be too slow to stop motion.
ZV