Shoot with your right arm as you lift your right leg. As you jump, imagine a string attached to your right arm and your right leg. Move them at the same time as you shoot, as though someone were pulling upward on the string. Your right knee should be bent and pointed toward the basket while your right arm moves up to shoot the ball. Arch your arm toward the basket. Shoot with your elbow slightly bent, so that your arm looks like the neck of a swan.
When you're doing a lay up, the shooting technique is slightly different from that of a regular shot. Instead of using your left hand to steady the ball, you want to shoot the ball using only your right hand. This gives you more reach, and since you're so close to the basket that it's difficult to miss the shot, you don't really need your left hand to steady the ball.
As you shoot, slightly rotate your wrist inward to give the ball a little bit of spin, rather than snapping your wrist forward as you would for a regular shot. The gentle spin will keep it from hammering off the rim or backboard with too much force.