In short, if you want to change your appearance, you must first meet an NPC called Trim. This character is a barber and can your character's current hairstyle or fur color by talking Trim.
You can meet Trim in his barbershop in Spurftop, and talking to him will also give a few side quests. One of which will unlock the ability to change your fur color, so go ahead and start it now.
Also, don't worry about any leg work, as all you will be doing is buying basically a pair of scissors from another NPC. Just follow the waypoint, and you will be fine and will unlock changing your fur color in no time! That said, when you finish the quest, you will be able to change your fur color for 400 leaves.
However, for more extreme changes such as fur color and appearance, you will need to seek out mutation zones. These are big pools of toxic waste, and you will find many of these in the area around the nuclear power plant.
There is also one easy one you can locate inside Bio Nucleus 6D, and after a small fight, you will be able to change your appearance via one of these pools inside.