This is a pretty common beginner question that arises in physics. The answer is that the equal and opposite force acts on a different object than the original force. When you are adding forces in a free-body diagram to find acceleration, you add all the forces that are applied to that object, you don't add forces that that object is applying to something else.
So, in the case of you pushing a box. You push on the box with a force of 1N, and friction acts on the box in the opposite direction with a force that's (presumably) less than 1N so the box moves forward. Those are the only two forces in consideration for the box. The box exerts a 1N force on you but that doesn't affect the motion of the box, that affects your motion. The reason you are able to move forward is that your feet are on the ground and there's friction between your shoes and the ground. When the box acts with 1N of force on you, the friction from the ground and you pushing off the ground with your legs act with force >1N on you and so you move forward with the box. And of course, there is an equal and opposite force that you are applying on the Earth through the ground, but the Earth is so massive that you don't change it's motion appreciably.