The moon rides on the same outermost ring as the sun. It exhibits the same characteristics as the sun regarding maximum height, rising and setting direction, and time above the horizon, according to where it is on the sun's ring.
The moon slips back like the sun does, but it slips much faster. Where it takes the sun a year to slip around the globe once, it takes the moon only a month. So instead of slipping back one degree per day like the sun, the moon slips back about 12 degrees making it rise almost an hour later each day.
Moon & Sun
Since the moon slips back more than the sun, the angle between the moon and sun are continually changing.
The shape of the moon changes with its angle with the sun.
Paths of the Full Moon
The Full Moon follows a path seasonally opposite the sun. At the Summer Solstice, the Full Moon behaves like the winter sun. Likewise, at the Winter Solstice, the Full Moon behaves like the Summer Sun.
Paths of the other phases
The Quarter Moon follows a path one season ahead or behind the sun. At the summer solstice, a waxing quarter moon takes the autumn equinox path, while a waning quarter moon takes the vernal equinox path. The crescent moon follows a path one or two months ahead or behind the sun, a gibbous moon is ahead or behind by 4 or 5 months.
The Moon travels on nearly the same path as the sun, but not quite. The moon's path is tilted relative to the sun's, resulting in the moon being either above or below the sun's path except when it is crossing. That is the reason we don't experience a solar eclipse at every new moon and a lunar eclipse at every full moon. Because eclipses can only occur when the moon's path crosses the sun's we call the sun's path "the ecliptic."