The Moon rises and sets every day, like the Sun. The Sun always rises in the morning and sets in the evening. The Moon, on the other hand, does it at a different time every day.

If you are watching the moon ...

Please note that after the full phase, the moon won't rise until after sunset and rises later each night. 

When it gets too late, get up just after sunrise and watch the moon continue east. 

When it gets close to crescent phase again, you may need to get up before sunrise to see earth shine again.

The moon rises and sets at specific times, according to what phase it is in:

The new moon rises and sets at approximately the same time as the sun.

The first quarter moon rises at mid-morning and sets at midnight. So it's at its height around dusk, not in the middle of the night.

The full moon rises at sunset and sets at sunrise. The full moon is the only moon that will be overhead in the middle of the night.

The last quarter moon rises around midnight and sets around mid-morning. So unless you're a late-owl, you've probably never even seen this moon.