According to Newton's third law, every action has an equal and opposite reaction. Why then does a box move when I push it? Shouldn't the two opposing forces cancel out to a zero net force?

The key to this question is realising that the action and corresponding reaction refer to forces that act on different objects. When I push the box, I exert a force on the box. As a result of this force, the box then also exerts an equal and opposite force on my hands. These two forces are not acting on the same object, as the force that I exert on the box influences the box's motion, and the force that the box exerts on my hands influences my hands' motion. These forces do not cancel because they influence the motion of different objects. Only forces that act on the same object can cancel to a zero net force. The force from my hands on the box can cancel out with other forces on the box, but not with forces on different objects (such as the force on my hands).