The Question:
At some point in your study of Newton's Laws you are going to have a question like:
"Okay, I understand that Newton's Third Law action/reaction forces between objects don't cancel, and they do influence the motion of the objects that they are exerted on, but internal forces are also Newton's Third Law action/reaction forces, but they do cancel, and they don't affect the motion of the object that contains them. What is the trick to splitting things up so that the internal and external forces come out right?"
The complete set of objects to be considered in a physical situation is called a mechanical system. However you decide to split up a problem into separate objects, Newton's Laws will work - there is no trick!
If you are concerned about the motion of your right hand, then you should consider your right hand to be a separate object. Forces exerted on it (by your left hand, say) then influence its motion. You can find the net force on your right hand and then use Newton's Second Law to find your right hand's acceleration.
However, if you are concerned about your motion as a whole person, then you should probably consider your hands as part of you - in which case the force exerted on your right hand by your left hand is an internal force, and does not affect the motion of your body as a whole. It seems pretty obvious once you get used to it...