As for designs, the possibilities are practically endless. What type of moon would you prefer? A more stoic image, like the Aztec moon; or something more cartoonish, like a half moon with a large grin and protruding nose?
Armband tattoos are a great option, each portion can depict the different phase in the moon cycle. You can also show the image of the moon in motion with stars trailing behind it.
A lot of people also like to use the image of the moon and sun in an intimate embrace. You can also use many fantasy figures such as fairies, dragons or one of the many moon goddesses (such as Diana, Mawu of Selene).
There are also many vintage and Victorian images that can be used for moon tattoos. Many of these feature a broad, round, content face. But there are several that feature cherubic angels and sweet young women lounging along the length of a crescent moon.
There are many moon tattoos to choose from, and a lot of room to get creative. Be sure that whatever you decide, it is something that will always has significance for you.