I want to make a group versus a page because I've been doing research and most sources say that pages are usually open to the public. The groups I want to set up must have some visibility but not so much that anyone can see updates and posts from the group.
I'm afraid you can't control post visibility individually using groups or pages. In a group you can set that, either everybody sees all of the content, or members only. For pages, most of the posts are public, but you can set, that only people from a specific country or speaking a specific language will see the post.
But you can do it as a user. So, may be you can create a user instead of a page?