They're cyclic, so which one comes "first" depends on where you define the start point. One easy way to remember it is that the lighted portion of the moon looks like the letters DOC in that order (with the new moon occuring at both the beginning and end of that sequence).
The full moon phase comes after the first quarter moon. The full moon appears as a complete circle in the sky, with the sun illuminating the entire side facing Earth.
The First Quarter moon
Waning crescent
Waxing Gibbous Moon!!!
The moon phase that comes after the first quarter moon is the waxing gibbous. During this phase, more than half of the moon's surface is illuminated, leading up to a full moon. The waxing gibbous occurs as the moon continues to orbit the Earth and reflects increasing sunlight.
Waning Gibbous is the moon phase that comes after the Full Moon.
The phase of the Moon that comes immediately before the new Moon phase is the waning crescent phase.
Waning Gibbous is the moon phase that comes after the Full Moon.
You get a full moon, then it will start to wain.
The new moon comes after the crescent moon
After the waxing crescent phase, the Moon enters the first quarter phase. During this phase, half of the Moon's surface is illuminated as seen from Earth, and it appears as a half-circle. The first quarter occurs roughly one week after the new moon, as the Moon continues to wax towards the full moon phase.
The waxing crescent is the first phase after the new moon.