The net force on the wagon is not zero. If the child pulls with a force F (where we set our coordinates so that "forward" is positive), then the force on the wagon is +F and the corresponding force on the child is -F. If these were the only forces operating (say, if the child is on a surface with ~zero friction, as if they were on ice skates), the wagon would accelerate forward, and the child would accelerate backward.

But these are not the only forces involved. In particular, the child is pushing on the ground. The ground exerts a force back on the child - let's call it A for "anchoring" - in response. So the total forces are:

• The wagon experiences a force +F from the child pulling.
• The child experiences a force -F from pulling on the wagon and a force +A from pushing on the ground.
• The ground experiences a force -A from being pushed on by the child.

These forces do sum to zero, satisfying Newton's third law, but the forces on each object do not (assuming F and A are not zero, the child is the only one who could be experiencing zero net force here if A and F are exactly equal).

The child is then relaying that reaction force to the ground.

The wagon accelerates in one direction and the Earth accelerates in the other.

Since the Earth is 24 orders of magnitude more massive, the F=ma acceleration of the earth is 24 orders of magnitude smaller.

Wagon goes 2mph east, earth goes 0.000000000000000000000002mph west.

Two equal and opposite forces exist, but they need not be applied to the same object. The wagon only experiences one force; that of the child pulling it. The opposite force is applied to the child, not to the wagon.

The child is pushing the ground back with his feet. So, forces acting on the child are balanced (so the child doesn't move back) but the wagon only gets one significant horizontal force (minus friction in the wheels) and doesn't accelerate.

If they were both floating in space, you'd have a situation similar to what you're thinking. Both the wagon and the child would accelerate towards each other. However, the wagon would move faster, because it (presumably) has less mass than the child, so the force accelerating the wagon would have more effect than the equal and opposite force accelerating the child.

They are not both floating in space. The child's feet are on the ground, and the wagon's wheels are on the ground. They are both accelerating towards the Earth's center of gravity. This means while the child experiences a horizontal force, that force also acts on their feet which are acting on the ground. Friction between the child's feet and the ground causes the child to stay in place. On the other side, the wagon's wheels do not produce as much friction and roll freely, so it moves. (Technically we could say the forces here have an impact on the Earth's motion, but in this case the difference in masses is so great we can ignore it.)

Imagine if you take the wheels off of the wagon and load it up with 400 pounds of concrete. Now something very different happens: the child pulls, but the wagon stays still. In fact, the child will very strongly feel the friction of their feet on the ground, and if they shift their weight just so they might even slide towards the wagon.

The force that the wagon pushes on the child is called "inertial force", it is simply the resistance to being accelerated instantly.

...by pulling the child back. The wagon is exerting force on the client to keep him from moving forward, and the child is exerting force on the wagon to make it move forward.

If they were on a frictionless surface, such as an ice rink, they would simply get pulled closer together. Since the child's feet are on the ground they have a lot of friction, so they don't move, whereas the wheels roll with little friction. If you replaced the wheels with cinder blocks, the wagon wouldn't move.

The child has unbalanced forcing acting on him too. The child is pulling the wagon with say 1 N, so the wagon’s pulling the child backward with 1 N too.

But the child is pushing his feet on the ground, so the ground is applying force of 2N on the child.

The child experiencing unbalanced forces and that’s why he accelerates.

We solve this with something called a force body diagram. The child, the Earth, and the wagon can all be diagramed independently.

The wagon has two forces acting on it. The force of the child pulling it, and the force of friction from the ground (maybe air resistance too, if you want). If the pulling force exceeds the friction force, the wagon accelerates.

The child has two forces acting on them. The wagon pulls back with the same force they pull on it (equal and opposite force that this question is asking about). Also, the ground pushes them forward with the same force they push on the ground while extending their legs (the normal force... Also a consequence of Newton's 3rd). If the normal force exceeds the reaction force of the wagon, the child accelerates forward (along with the wagon).

The ground has two forces. The friction with the wagon actually acts forwards on the ground since it works backwards on the wagon. And also the force of the child extending their legs backwards also pushes the ground backwards. We already know that the force from the kid's legs exceeds the reaction force of the wagon in their hand. And we also know them at this pulling force exceeds the friction of the wagon to the ground. Therefore, the kid's legs must be pushing harder than the friction and the Earth must accelerate backwards for the kid to be able to accelerate forwards.

The thing is, the Earth is massive. It will not accelerate very much. Negligibly in fact. So basically, we can if ore it and only focus on the child accelerating.

Edit: crude diagram the forces acting on the cart are in green. The forces acting on the child are in blue. The forces acting on the Earth are in red. Every force has an equal and opposite pair. F for friction. P for pulling. R for running all show up twice in opposite directions (and equal magnitude) and their relation tells you which direction each object is accelerating.

Not an equal and opposite force, an equal and opposite reaction. The wagon does put an opposite force on the child, but can only put so much force into this. If the child is strong enough to overcome that force, then the wagon must make up the difference by reacting in some additional way, and that is where the motion comes from.

Here is another way to think about Newton's 3rd Law.

The sum of forces in a state of equilibrium are zero. Essentially if an object is not moving horizontally, there are either no horizontal forces, or the forces sum to zero. In this case, the wagon accelerates because a horizontal force was applied (the child).

The child applies a force on the wagon, and consequently, the wagon does pull back on the child.

Now let's consider the wheels. There is a now a horizontal force applied to the wheels. Consequently, the ground applies a negative horizontal force (Friction). The wagon will only move when the force applied by the child moves is greater than the friction force.

An easier example may be to imagine a child and the wagon on a smooth (frictionless) surface, such as a sheet of ice. If the child pushed on the wagon, the wagon would push back onto the child, and both would move apart.