Hi @William Miller - Welcome to Salesforce dev and thank you for trying out Code Builder.  I'd love to hear your feedback on how it was to get started as someone new to coding.  Feel free to share here or drop me a direct message.

Where are you looking to store your work?  In another org?  In source control?  Or as local files on your device?

If you're storing them in another org, you can connect to that org from Code Builder and run a deploy command.  Here's a link to documentation on connecting an org and example steps to deploy different objects are listed in sections below.

If you want to use source control, git is baked into the editor.  You can see information from Microsoft on getting started with git here. 

Another option is to save your workspace as a local file.  You can do that by clicking the icon in the top left of Code Builder (three horizontal lines) --> File --> Save Workspace As.  In the prompt that comes up, there's a button labeled 'Show local'.  If you click that, you can see your local file system and save it there.  (assuming you have the permissions on your machine)