How do a planet's moons and a planet's rings interact? I am making a video game which features a planet and its moon. And I thought I would add rings too.

But that got me wondering... how do a planet's moon and a planet's ring influence each other?

Does a moon clear out all the debris in its orbit? In other words: does it make a gap in the ring? If so, how large is the gap in relation to the moon's radius?

If not, does the moon travel along as all the other matter in the ring? Would it not make the debris fall to the moon, thereby growing the moon?

Lastly, are both the rings and moons typically in the same plane?
First, there are different types of rings. Using saturn as an example: there are icy particles, dust bands and more. These interact differently if they were to hit a moon, for example enceladus.

Every little particle in a ring is a „mini-moon“ and obeys the same laws (see kepler) as the big ones. 
They technically all have their own orbit.

The main rings of saturn have a high enough particle density to exert gravitational influence on each other.

There are hypothesis that suggest that rings are made of shattered moons (like through collision) or another one that says that the rings are made of particles that „failed“ to make a moon in the first place.

So now, what happens when rings collide with moons is dependent on the type of ring.
Enceladus is a small icy moon in the so called E-Ring of saturn. This ring is small and made of water/ice, just like the moon. Because the ring is so very small and such a water/icy „cloud“ would  not really stay together, it is believed that it is „refilled“ by enceladus constantly.
The surface of enceladus was observed to be smooth, not really having craters one might expect, but this could also be due to geological activity. Enceladus also „geysirs“ out water all the time, so this might be how the ring is kept. 

Edit: I recommend checking out this link: https://courses.lumenlearning.com/suny-astronomy/chapter/planetary-rings/