Short answer is easy. Long answer, there's lots of maths involved, and I'm out of practice with my maths, but there's a few basic parts that your students should be able to follow. It's a little confusing, not too bad.
We can ignore the Earth's motion around the sun for the most part but not the Earth's escape velocity.
Start with Newton's cannonball thought experiment. That will get you into space, and that's an example of orbital velocity. With a fast enough launch, the cannonball enters into and stays in Earth's orbit.
2nd, explain the difference between escape velocity and orbital velocity. Escape velocity is greater than orbital velocity and the cannonball that orbits the earth hasn't really escaped the earth, it's still in the Earth's gravitational sphere of influence. You need escape velocity (or very close to it) to get to the moon, about 11.2 KM/s. You can get away with a bit less cause the Moon hasn't fully escaped the Earth either.
Astronauts can't get shot out of a cannon at high speeds cause it's too many G forces, so what rocket launches do is accelerate over a few minutes. (you could probably look up the precise amount of G forces), but within G forces that humans can tolerate. If they experience 4Gs, they're accelerating the equivalent of 3Gs (or 29.4 meters per second), at least at first, as they move away from the Earth, the gravitational acceleration towards the earth drops and after only a few minutes they can shut down the launch and drift towards the moon. Most of the trip to the moon is done in this way, just drifting through space. You don't want to go too fast cause you'd just need to slow down as you approach the moon again and the moon has no atmosphere to assist the deceleration like the Earth does.
The ships momentum traves through the combined gravitational field of, primarily, the Earth, Sun and Moon, and the Maths get pretty complicated, so the the launch needs to be quite precise.
The Moon also orbits the Earth at about 2,300 miles per hour and that orbital speed has to be matched as well as, since the rocket will often enter a partial orbit around the moon, that orbital velocity around the Earth needs to be matched too.
Once this is done, landing on the moon requires a simple enough deceleration to have a gentle enough touchdown to not harm any equipment. (this probably isn't as easy as it sounds).
An orbit around the moon isn't strictly necessary but a partial orbit was used to assist landing on the desired spot. A partial orbit around the Earth was also used in the launch to the moon.
They accelerate away from the Earth in 2 different times, first to leave the earth and get into Earth orbit. That's the big one, then from a different spot (and not even an hour later) to accelerate out of Earth's orbit towards the Moon.
See diagram.
At least, that's a pretty basic summary. I welcome correction if I got anything wrong or missed anything important.
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