Once you have your moonshine there are an infinite number of things you can do with it. I am going to cover only a few of the more common ones.
This is the process of literally watering down the concentration of alcohol. The primary purpose of this is to add volume to alcohol. For example: 1 quart of 160 proof moonshine can be watered down to 2 quarts of still very potent 80 proof moonshine.
This is the process of further increasing the proof of an already distilled moonshine.
Carbon is used much like a water filter to remove bad tasting contaminates from moonshine. Unfortunately, it also removes the good tasting flavors as well. Because of this they are normally used to make a neutral moonshine that will then be mixed with fruits or wines later.
This is the process of simply adding flavors and/or sugar to a jar of moonshine to enhance the taste. From apple-pie to coffee, nearly everything can be used. Use a coffee filter to strain the mess after letting the concoction sit for a few weeks.
Many types of liquor have a special ageing process that defines them, one example is Whiskey. Part of the process is that it is stored inside a charred-oak barrel for a specified amount of time. Since most beginner moonshiners do not have access to oak barrels this can be recreated by simply charring a piece of white oak and putting it into a mason jar filled with moonshine. Over time the moonshine will age, turn color, and become a very basic whiskey.