The best way to find out how quick a camera can snap a photo from button press to final image appearing on the LCD screen is to test it out before you buy it, or check reviews to see which camera is quickest. Ideally, a compact camera will have a shutter lag of less than 0.5 second to keep up with fast-moving subjects. A digital SLR will be even faster, offering near-instantaneous image capture.
If you already have your camera and want to find out how fast it is, make sure your camera is ready to take an image, press the shutter button, and time how long it takes for the image to appear on the screen.
If your shutter lag is particularly long, you can pre-focus to help lessen the delay. Pre-focusing is a technique where you depress your shutter button halfway before you get ready to shoot your subject in action. This means you point directly at your object while at the same time pressing the shutter release button down halfway. Doing so, you can save time and snap your picture much faster as you are already halfway through the process.