I'm into development on this website, although it's not finished, many hours have gone into constructing it. The website is not live yet, but I am noticing a few weird things here and there. I pulled up your website and noticed that there were a ton of changes after I purchased it sometime in Feb. - March (2017). I would like to update to the newest version of the theme, but I am a little hesitant since I have not have to update an entire theme like this before. What do I need to do to make sure I don't loose anything in the process?
As long as you didn't do any direct template editing you should be totally fine to update via FTP or through a plugin that helps with the process. It is always recommended that you use a child theme as well, there is one included with the download package that you could install.
I don't think I have. Can you give an example of changing it?
You would know, you would need to go through FTP or the theme editor. It's how you edit the theme files at a code level. You should be good to update to the latest version, you can use this plugin which provides a nice GUI for replacing the current version with the new one: https://wordpress.org/plugins/easy-theme-and-plugin-upgrades/