So some time passed and I was no longer running the most current version of Moodle; and then there were the security warnings that made me think I should update my site. I've put it off for as long as I could. It was time for me to figure it out anyway.
Reading the directions online about how to update your version of Moodle kind of freaked me out. I followed it along pretty well until it got the part about backing up my database and then it lost me completely. When I finally had the motivation I made a few mistakes, but have discovered a very simple way to do this for anyone else who is a beginner like me.
1. Log into your GoDaddy account and get to your site management menu. From there you can back up your database. You can also make a copy of your moodle and moodledata folders.
2. Go to the administration block of your Moodle site and turn on maintenance mode so that users know the site is unavailable. It will also prohibit them from making any changes.
3. Download the latest version of Moodle from moodle.org.
4. Copy your config.php file from your current version of Moodle and your .htaccess file if you created one.
5. Take any folders that you have added to your Moodle folder and put them in the new version of Moodle (any themes that you have added; portfolios; spell check; etc.).
6. Rename your Moodle folder in your GoDaddy account (most of these steps are for your protection in case something goes horribly wrong).
7. Upload the zipped Moodle folder to your GoDaddy site.
8. From here you can unzip the file in the File Manager.
9. Replace the config file to overwrite the new one that would have you go through the installation process again.
10. When you go to your Moodle site you should be prompted to update Moodle. If it doesn't then try it again as something horribly wrong has happened.
11. Update Moodle and delete any extra back-ups you don't need for your website anymore (I'd save them somewhere just in case you need them in the future).
Do not be intimidated by this process. It is much easier than it sounds and most of the steps are just backing things up. It really only takes a few minutes.