Your choice for the home page is either a static page, which never changes, or a blog page, which displays a list of excerpts from posts that you create, in decreasing order by date. Judging by what you want to do, I would suggest using a blog page for your home page, which is the default setting for WordPress. Then you could create a post for each chapter every so often, and the excerpt for the latest chapter would appear at the top of your home page.
For example, you might create a post for your first chapter, then wait a few days, then create a post for your second chapter, and your home page would show excerpts from both chapters on it, with the second chapter’s excerpt at the top. If your visitors were interested in reading the full chapter, they would click on the title for that post and be taken to a new page where they could read it.
To figure out what theme you are using, go to Appearance > Themes from your admin dashboard, and the name of your theme should be at the top.