Dear student
The action and reaction forces DO NOT cancel out.
Because action and reaction pair never act on the same body,they always act of the different body.
In this case when a person is hitting a ball it is applying force on ball(action) and ball is applying back that equal and opposite force(reaction),so the forces acting on the ball are only the action force from the person(and obviously gravity) and so the ball moves.The motion of a body is decided by the forces acting on the body not the forces which the body is applying on some other body.
Regards