should I download from appstore or apple support?

If you need a full installer for an operating system you need to use the AppStore, the Apple Support Downloads Site contains updates to OS's, these updates require you to already be running an earlier version of the OS you want to install the update on.

So in summary if you want to install a new OS (that is called upgrading) you need to download the full installer, and that can only be downloaded via a link to the App Store.

Use Safari to connect to these links.

How to upgrade to macOS Sierra - Apple Support

How to upgrade to macOS High Sierra - Apple Support

How to upgrade to macOS Mojave - Apple Support

How to upgrade to macOS Catalina – Apple Support

If you for example are already running macOS Sierra or other and want to update it to a later version of the same OS that

is called an update. Updates for Sierra and High Sierra can be done through the App Store Updates page or from the Apple Downloads Support Site. Updates for Mojave and Catalina are done through System Preferences> Software Update

or the Apple Downloads Support Site.

should I upload my operating system

No, you shouldn’t upload something that isn’t yours to upload. Perhaps you meant ‘download’?

from the google support site or the appstore?

Don’t get Mac software from the Google Support site. The Apple Support site is fine, though.

There just seems to be a difference in how they do it?

What difference do you think you detected?

How to upgrade to macOS Sierra - Apple Support

If your computer's hardware meets the requirements for the upgrade, you can download the upgrade either from the Apple Support site (like the link posted by Urquhart1244) or on the App Store.