The 1st May full moon happens on May 1. It'll look full on the nights before and after that, too. The 2nd May full moon - a Blue Moon - will fall on May 30-31.
The full moon, opposite the sun in its orbit, shows us its sunlit side. The moon appears full a couple of days before and after the exact moment of full moon.
A waning crescent moon can be found in the east before sunrise. It's waning toward new moon, that's when the moon will be between the Earth and the sun.
The first quarter moon is halfway between new and full moon. Half of the moon is illuminated and it's a quarter of the way through it's orbit around the Earth.
A waxing gibbous moon is in the sky before it's dark and lights up the early evening. It's more than half lighted, but not quite full. It sets after midnight.