Hi there, there are two options to update macOS: 1) Using System Preferences > Software Update (automatic), or 2) Manually, by creating a bootable USB drive (more details on that here: How to create a bootable installer for macOS - Apple Support).
The first option won't allow you to upgrade to macOS 11.x.x (i.e. Big Sur), because a newer version is now the current one (12.x.x or Monterey). So, I'd recommend to upgrade to Big Sur manually (if that's what you're after).
Once you upgrade to newer macOS, it's simple to get Logic: just log into App Store with your Apple ID (if you don't have one, create it first) and then browse to Logic Pro page where you can make the purchase. The app installation is done automatically through App Store as well.
That's it in a nutshell, but please don't hesitate to get back if you have more questions. Cheers!