It gradually drifts through the sky relative to the sun. At full moon, the moon is opposite the sun, and rises when the sun sets, and sets when the sun rises.
Since the sun rises in the East and sets in the West, that means the Earth is spinning towards the East (towards the sun as it rises).
The Moon orbits in the same direction the Earth spins, but more slowly.
So the Moon gradually drifts Eastward, further behind the sun as it rises, by 1/28 of a full circle every day. So every day it rises a bit later, by a little under an hour every day..
So starting from new moon we have:
New moon rises as sun rises, sets as it sets
First quarter moon rises when sun is at zenith (midday) sets at middnight
Full moon rises as sun sets, sets as it rises.
Third quarter rises at midnight, sets at midday.
You can look up the moon rise and set times for every day for your part of the world here:
Note that the new moon and full moon can happen at any time of day and night, so when you see it says full moon for a particular day - that doesn't mean the moon rises as full moon, it might be full moon any time of the day or night.