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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To understand this, you have to think about how the earth and moon move relative to the sun (think or a model of the solar system).
Just like the earth has an orbit around the sun, the earth’s moon has an orbit around the earth. While the earth orbits around the sun in about 365 days (a year), the moon orbits around the earth in about 27.32 days.
We see the moon because of the light from the sun reflecting off of it. The phases of the moon that we see are a result of the moon, the earth, and the sun being in different places relative to each other (the moon along its orbit around the earth and the earth being along its orbit around the sun). The cycle of the moon’s phases is about 29.53 days. This is the time it takes the moon to make one revolution relative to the sun (it’s not the same as the time it takes the moon to orbit the earth because the earth is also orbiting the sun).
https://moon.nasa.gov/moon-in-motion/phases-eclipses-supermoons/overview
https://solarsystem.nasa.gov/moons/earths-moon/lunar-phases-and-eclipses/
This video is really helpful in visualizing this: https://youtu.be/wz01pTvuMa0
The Moon rotates around the Earth pretty slowly – one rotation every 29.5 days. This means that sometimes the Moon is on the same side of Earth and the Sun (that’s a new moon, by the way), so it rises with the Sun and sets with the Sun. Two weeks later, the Moon is on the side of the Earth opposite the Sun (that’s a full moon) – so it rises when the Sun sets, and sets when the Sun rises. Then of course there’s everything in between, so the Moon might rise at noon and set at midnight, or vice versa.
Both sun and moon rise in the east and set in the west once per day. This is because they’re both essentially staying still from our point of view as the Earth’s spins around its axis once every day. We obviously also orbit the sun and the moon orbits us but those things take a year and a month compared to a day it takes the Earth to spin around its axis.
Moonrise and set each day happens about 49 minutes later from one day to the next, on average (it can be a little more or less for any given day). Other answers here explain the mechanics of why that happens pretty well.
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.
During a full moon, the moon is on the opposite side of the earth from the sun, so we see its face fully lit. Since it’s on the opposite side of the earth, we’ll see it on the opposite side of the sky: the full moon rises in the east as the sun sets in the west, and vice versa.
During a new moon, the moon is on the same side of the Earth as the sun, so the unlit side faces us. That means that it would appear right next to the Sun in the sky (if we could see it), so it rises as the sun rises, and set as the sun sets.
In general, the Moon orbits the Earth, completing one full circuit from new moon to new moon every 29.5 days. So every every day it moves a little farther in its orbit and appears a little farther to the east in the sky, and every day it rises and sets about 50 minutes later than the day before.
So, for example, the first quarter moon rises and sets about 6 hours after the Sun, the third quarter moon is about 18 hours behind the sun, and so on.
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