Regardless of how shaky your hands are, shutter lag does not cause blurry images.
The following techniques will minimize the blur that is caused by your own movement.
When taking a picture, steady the camera by doing the following:
Stand with your feet shoulder length apart and one foot slightly in front of the other, or lean against a wall (or something sturdy).
Put the camera up to your eye, look through the viewfinder.
Cradle the bottom of the camera/lens with one hand.
Tuck your elbows in close to your body.
In one smooth motion press and hold the shutter button until the camera is completely done taking the picture.
Do use the LCD screen on the back of the camera to compose your shot (only use it to review your shots after you take them), do not hold the camera straight out in front of you, do not jab at the button when taking the picture.
If your subject is moving, you'll might need to increase the shutter speed . You'll have to look at your camera's manual to see how to do that, but most cameras have a shutter-priority mode that is simple to use. Point-and-shoot cameras are notorious for setting shutter speed too slow in full automatic mode. They often set the shutter speed at the inverse of the focal length, thinkig that should be enough to offset the effects of hand-held camera shake. For instance, at 50mm-equivalent zoom the camera may set the shutter speed to 1/60 sec. For the average snapshooter, 1/60 may be too slow; 1/125 may be more appropriate, and with moving subjects (or if the photographer is on a moving platform, like a ride at WDW) 1/250 or faster would be even better. Of course, using a faster shutter speed means decreasing the length of time that light falls on the sensor or film, so you'd have to also admit more light by using a larger aperture or increase sensitivity by using a higer ISO.
Some cameras also have a "burst mode" or "continuous mode", where the camera will take several pictures in succession as long as you hold down the shutter button. Don't bother using this mode with flash, because your flash will need time to recycle between each shot, defeating the purpose of this mode.