This came up on chat, so we figured it would be good to outline it in the Forums. While in the most general of terms the moon rises and sets in the East/West there's patterns astronomically to consider, as well as chronology.
First, let's consider lunar phases in relationship to time of day.
Moon phase Moonrise Moonset
New Sunrise Sunset
1st quarter Local noon Local midnight
Full Sunset Sunrise
3rd quarter Local midnight Local noon
Then, Moonrise and Moonset:
The chart is set for Northern Hemisphere, for Southern Hemisphere replace North with South below.
Season Postion of Moonrise/set
NEW 1st FULL 3rd
Winter SE/SW E/W NE/NW E/W
Spring E/W NE/NW E/W SE/SW
Summer NE/NW E/W SE/SW E/W
Autumn E/W SE/SW E/W NE/NW
Like the Sunrise/set positions, the amount of variation depends on your lattitude.