Getting started in becoming a Moon watcher is easy. Next time you can see the Moon, stand outside and start watching it. Study it in real detail see what formations you can pick out, try to find anything that you can. Gazing at the Moon with the naked eye is the best way to start watching the Moon.
Once you have begun watching for a while, try purchasing a good pair of binoculars or a telescope. That will enhance your pleasure greatly as you watch the Moon pass through the night sky. You will be able to see larger images in greater detail, and truly begin your study of the Moon in earnest.