In order to get this achievement, you have to complete the game without changing the date on your computer. Normally you would have to wait for both a full moon and a new moon in real time, however there is an easy way is to do it in one setting by using the in game moon grotto.
(Note that this method is only available if you have not yet gotten the Dark Moon Trigon, as in the process of obtaining the Trigon you destroy the tree the key is located in.)
When you access the dream world when it is not a Full or New moon, you will see a bear type creature. On your second or later visit, you can follow him to the shore. If you go to the coastline, you will get a message on screen that says 'Believe'. Just keep going right into the water and you will pass over to the next area. Continue following him and he will lead you inside a tree and start a battle. This battle is nearly identical to the shadow creature from the start of the game. Once defeated he will drop a key.
You can use this key in the outside world to access the moon grotto. The moon grotto is located several screens to the right of the cabin, just before the screen with the cliff top where you raised your sword to extend a bridge. You will see a small path headed south to a door in the lower right corner of the screen. Once inside you can sit on the center pedestal to change the moon phase. Quitting the game or leaving a dream will revert the moon to the 'real world' state, so you will have to change it each time.