When I had my consulting business, the website included a post about Newton's Laws and how they apply to rockets. The business and the website are history, but people really seemed to like that post, so it deserves to be revisited here.
What prompted the post was the many confusing and/or wrong websites claiming to explain how a rocket can be propelled in the vacuum of space. They typically go like this:
1. Rockets produce thrust by expelling hot gases out their nozzles.
2. Newton's Third Law explains why a rocket can accelerate in space, where there is nothing to push against.
Some really bad sites will say that the expelled gases are the action, and the forward motion of the rocket is the reaction. This is way off, because "action" in this law means force, not motion or momentum. You may think this is trivial, but anyone who doesn't know the difference between force and motion is going to have a rough go in physics class.
Additional note for geeks: Not only does the Third Law not explain why a rocket works, there are electric rockets that are in strict violation of the Third Law but work just fine
3. Well, maybe it's Newton's Second Law that explains it.
Not really. The Second Law does talk about motion, implicitly as momentum. Momentum is mass times velocity. The Second Law says, The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed. But what is a body? It's a fixed quantity of matter. There is no fixed quantity of matter that corresponds to what we think of as the rocket. The best definition of "the rocket" is "the stuff the engine has to push forward." That includes both the airframe and the fuel it holds. But at the next instant, some of that fuel has been burned and blown out the nozzle, and is no longer part of the rocket. The rocket's mass is rapidly decreasing, so it isn't a body and hence is not described by the Second Law.
The fact that the rocket isn't a body also explains why the Third Law has little to do with the total thrust. If you draw an imaginary boundary around the rocket, no matter where you draw that boundary, somewhere, fuel or exhaust is pouring across the boundary. That fuel or exhaust exerts pressure on the stuff on the other side of the boundary, which accounts for a small part of the thrust. But the vast majority of the thrust comes from the fact that mass is crossing the boundary.
To understand the motion of a changing quantity of matter (an "open system"), we need a generalization of the Second Law that didn't come along until some time after Newton. He was mainly thinking in terms of planets, which are about as close to the strict definition of a body as you can get.
It is true that Newton phrased the Second Law in terms of momentum, so some people think it accounts for momentum change due to mass change as well as velocity change. But it says, and means, that it applies only to bodies.
You may have noticed that one of the websites I called out above is a NASA site! Here's one that is just horrible; they have an equation at the end that is plain wrong. I am going to resist the urge to make a NASA joke here. I know a lot of people at NASA and they're all pretty sharp. My guess is that those websites were made for NASA by "education contractors" that were chosen based on some criteria other than technical competence. NASA ought to have some of their real rocket scientists review them.
If you want the equations in all their detail, check out this site from MIT. Notice that the word Newton never appears on that page.
In conclusion, any explanation of rocket motion in terms of Newton's Laws is going to turn into a mess, because in fact they are insufficient by themselves to do the job. Because rockets change their mass, they're probably the worst possible way to try to introduce the laws of motion to a beginner.