The Moon doesn’t necessarily rise at sunset and go down at sunrise, sometimes it’s up during the day. So what does it do? Are those times varying depending on it’s phase? Is it random? Are they always the same for a same phase? Do they vary across the year like the Sun’s do?
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The times of Moonrise and Moonset depend almost entirely on the Moon’s phase. A full Moon rises around 6 PM and sets around 6 AM. A half Moon rises around noon and sets around midnight, or rises around midnight and sets around noon, depending on which side of the Moon is lit. A new Moon isn’t really visible as the Sun is only lighting it from behind, but if you could see it you would find that it rises around 6 AM and sets around 6 PM. A crescent Moon rises and sets just before or after these times (again, depending on which side is lit).
So why is this? Well, consider what it takes for the Moon to be in a certain phase. For the Moon to be full, it needs to be exactly opposite the Sun. That is, you, the viewer, will be (almost exactly) on a straight line between the Moon and the Sun. Of course, it can’t be an exact straight line because that results in a lunar eclipse (and indeed, lunar eclipses only occur during a full Moon). But there’s a little bit of wiggle room in the orbits that means that most of the time, the Earth doesn’t get in the way of the Sun’s rays reaching the Moon.
The other thing you need to realize is that the Moon’s orbit around the Earth is much slower than the Earth’s rotation around its own axis. The latter of course takes a day to complete, whereas the Moon takes about 4 weeks to go around the Earth. So, over the course of a day, we can consider the Moon’s position relative to the Earth to be pretty much fixed. Imagine a stationary marble with next to it a spinning tennis ball.
When the Moon is full, it has to be on the Earth’s “night side”. That’s the only way for it to catch the Sun’s light across it’s entire surface. So that’s why a full Moon rises when the night begins and sets when the night ends (on average; the actual times of Sunset and Sunrise on any particular day vary with the seasons of course).
A half Moon is catching sunlight only from one side. So for that to happen, the Moon, Earth and Sun have to make a right triangle, with the Moon on the end of the shorter leg. On a clock face, the Sun would be at 12, the earth in the center and the Moon at 3 or 9 (but do not confuse these clock-face positions with the time of day that the Moon is visible). This puts the Moon either over the Earth’s evening or morning side, i.e. directly over wherever on the Earth it is 6 AM or 6 PM, depending on which side of the Moon is lit. Meaning that it will be highest in the sky at those times, but also visible 6 starting 6 hours earlier and ending 6 hours later.
A new Moon is when the Earth, Moon and Sun are again arranged in (almost) a straight line, but now with the Moon between the Earth and Sun. Occasionally this results in a solar eclipse, but not usually, again due to wiggle room in the orbits making it so they don’t usually line up perfectly. This puts the Moon on the day side of the Earth, which means the side of the Moon that faces the Earth now gets no sunlight at all – all the light hits the opposite side of the Moon.
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