In Neo-Paganism the moon cycles are often seen as a way to think about planting/growing/nurturing (during the waxing moon) and letting go/banishing/dcreasing (during the waning moon).
Folklore certainly has lots of advice on planting and harvesting related to moon phases.
The full moon is full power. And the new moon is a time for spell casting–what do you want to bring into your life over the next month.
The dark moon is the one to three days prior to the new moon being visible and is a time for banishing. (Makes sense–since the moon itself has “vanished”.)
In the Paganism 101 curriculum, we learn about the phases of the moon and correlation to pagan rituals and practices in session 9.
Here’s a fun activity to do with pagans and/or kids.
Photo from: https://4thgradefrolics.blogspot.com/2012/05/howling-at-moon-and-happy-happy-joy.html