I started to work on my Wiki that is based off of my novels. Though I don't want to reveal the majority of my original characters, art, and my written articles to the public, as stated in the header, is it possible to set my Wiki on private? Or at least maybe a different Wiki server that provides private Wikis?
No, I'm afraid such private wikis can't be created on Fandom. I'm not well-aware with other wiki-hosting services but I've heard of Miraheze. You can check it out and see if it's sufficient for your need. :)
If you want a truly private wiki, you can always try hosting your own. It's quite a bit of work, though. I'm guessing you don't want a truly private wiki, because even Miraheze wikis would probably be view-able by the general public.
I can't recall Miraheze's policy but they at least allow private editing. However, from what I can tell, Miraheze is run by a group of "volunteers" so I don't know how much support you will get. You can use ShoutWiki which is "professionally run" and offer completely private wikis, I think at no cost. There is a paid tier system but I believe that is related to support and features, not the type of wiki you can create.