I think you mean shutter lag (or click-to-capture time), the time from the moment you press the shutter button to the time the camera actually takes the picture.
You can significantly reduce shutter lag by half-pressing the shutter button when you've framed your shot, then fully press it when you need to. Lag time should be much shorter, though still not instant.
I recommend the Kodak EasyShare DX7630 or the Sony DSC-P150 for indoor shooting; if you don't care about a weak flash, the Sony DSC-T1 will suffice. The Konica Minolta DiMage Z3 is fast as well and comes with image stabilization, but it doesn't do so well in low light and is not considered a "compact" camera.
I have an Olympus Camedia C-3040ZOOM which I received as a gift. There are 3.3 megapixels, 3x optical zoom, 2.5 x digital zoom, f 1.8 lens. (I would probably upgrade the megapixels if buying a new one, but am generally happy with everything other than speed.)
Anyway, the camera works fine for still subjects, but the shutter speed is WAAAAY to slow to get good pictures of my running, jumping, always moving toddler. I shoot when he smiles, but by the time the camera snaps the photo his eyes are closed and he's already run away!
Can anyone recommend a great camera that will snap photos as soon as I press the button? How can I find out about shutter speed on each model? I couldn't find it in the reviews. (I am technologically illiterate, so forgive me if this should be obvious...) Thank you!