I deployed to got, but then whenever I change any code and push it to the gh-pages branch via command line it doesn’t change the actual site itself. When I check the repository after the push, it has all of the new code, but even then the site does not change.
Yes, any pushes to the branch you selected in your repository settings, whether done on GitHub.com or on your local machine, should automatically trigger an update to your site. The build process for a Pages site can sometimes take around 10 minutes to complete so you may have to wait for a little while before you see the latest version of your site.
If you’re still not seeing changes to your site after waiting for this time I’d recommend checking which branch your GitHub Pages site is set to build from, and make sure you’re pushing updates to that branch.