Now you know why Newton's Third Law action and reaction forces don't cancel - it seems pretty obvious once you get it. The problem is that sometimes Newton's Third Law action and reaction forces DO cancel...
Have you ever noticed that you can't push yourself? You can push a book and it accelerates, and you can push another person and they accelerate, but you can't accelerate yourself by pushing yourself.
You can lift a book off the table, and you can lift another person off the ground, but you can't lift yourself off the ground. (A person can't literally "pull themselves up by the bootstraps" as the old saying says...)
Suppose that one part of an object is pushing on another part - your right hand is pushing on your left hand. Newton's Third Law tells you that both hands exert forces, and that the force on your right hand is equal and opposite to the force on your left. Previously, you saw that the force that your right hand exerts on your left hand accelerates your left hand, and that the force your left hand exerts on your right hand accelerates your right hand - and you can see and feel that happening.
Notice, though, that no matter how hard you push, the forces your hands exert on one another will not accelerate your body as a whole.
Forces exerted by one part of an object on another part of the same object are called internal forces - and
internal forces never influence the motion of an object.
Newton's Third Law action/reaction forces between objects do not cancel - but internal forces (Newton's Third Law action/reaction forces within an object) do cancel.
Forces between distinct, separate objects are called external forces, and external forces DO influence the motion of objects.