An excellent question!
You are correct in stating Newton's third law of motion that every action as an equal and opposite reaction.

Newton's third law is a law of the forces acting on an object, which are the product of mass and acceleration.

So let's take a closer look at your bullet example.

Initially the bullet and gun are in contact with each other, so we consider them as a single closed system. When the gun is fired, a combustion reaction heats up the air causing it to expand rapidly and shooting the bullet.

We can recast Newton's third law from forces to change in momentum. Momentum is defined as mass times velocity.

Before the gun is fired, everything is at rest. The bullet experiences a change in momentum that must be matched by a change in momentum of the gun. This is Newton's third law in action. The change in momentum of the gun is called recoil. Because the gun is much more massive compared to the bullet, it does not recoil as noticeably. You can see this whenever someone shoots a gun; they need to ground themselves to hedge against the recoil.

You can apply the same principle to how a rocket moves in space. The fuel that is combusting at the tail-end of the rocket is what is pushing on the rocket and propelling it forward (the same is true when the rocket is being launched, unlike the misconception that the rocket pushes off the ground).

You have almost stated Newton's Third Law correctly! What you forgot is that the action is applied from object A onto object B and the reaction arises from object B to A! So, what we refer to as "action-reaction" is a pair of forces that is applied to two objects, not one, and this is why they cannot cancel.

Let's think of two examples to convince you. Let's assume you play pool (billiards) and all the balls have the same mass. And let's say I hit a striped ball on a solid one. Initially, the striped ball is moving, and the solid is stationary. But after the collision, the striped will be stationary and the solid will move. How did the solid ball move? It moved because the striped ball applied a force on it, in Medieval terms called "an action". How did the striped ball stop? It stopped moving because the solid ball applied a force on it, in Medieval terms, "a reaction".

"Action" and "reaction" are nothing more than two forces. But Newton gave them this name because they are special forces, they have same magnitude and opposite directions (and are applied at different centers of mass). And this is the reason why if the two billiard balls have the same mass, the striped ball will stop immediately and the solid will move with the velocity that the striped had, assuming the collision happened at the center line. If not, new physics of rotation had to be discovered!

Now, the second example, is more complex in reality but may help you to think of "action-reaction" pairs is easy. Imagine you are taking your dog (or other favorite animal) for a walk, so there is a leash in between you. The leash can only sustain tension, not compression. You pull the dog towards you because you want to keep moving (you perform the action). But the dog resists (reacts) and pulls towards its favorite tree / street corner. The dog experiences your force (action) via the leash, and you experience its force (reaction). But these forces can only be the same in magnitude! But you are definitely getting pulled by the leash (and the dog) and only the friction of the ground and your muscle power helps you to stay put and not fall ! So, definitely, you see that you who applies the action, experiences a force that is not being cancelled.

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The gun moves, too, but in the opposite direction and with the opposite force. If you ever fire a gun, you will learn this, as the gun will have something called a "kick", and the more powerful the gun, the more powerful the kick. This kick is the movement of the gun backward as a consequence of propelling the bullet forward.

The thing you should be aware of: the opposite reaction is not applied to the same object. If you treat the two objects (or more) as a whole system, then the opposite reactions will get cancelled. But individually, object A exerts force F to object B; object B will exert the opposite force F to A (not to B). So individually, the total force on A or B is not balanced! As for the gun, it is the same principle. The explosive chemicals will expel the bullet out, at the same time, you will also feel the backlash (the opposite force). That's why you have to hold the gun firmly while opening fire. If no one is holding the gun, then the bullet will be going forward and the gun will move backward.

It’s great that you’re really trying to use you’re understanding of Newton’s laws to explain everyday phenomenon, but you’re missing a key idea. When the gun fires, it pushes the bullet in one direction and the gun in the other. Both the bullet and the gun experience the same force but in opposite directions. These two forces are equal and opposite of each other. This opposite force manifests itself as recoil, which is where the gun pushes back whenever it is fired. You may be wondering why the bullet travels so much faster if both the gun and bullet experience the same force. This is because the heavier the object is, the less velocity it will experience when forced. The exact numbers are a bit more complicated than that but the basic idea is there. Thank you for your question!