How can I upgrade High Sierra to Mojave or Catalina?

Apple prefers folks be running the current release.

Running older versions involves increasing effort on the user to prepare and to maintain that configuration.

For the older upgrades, see > How to get old versions of macOS - Apple Support

Download the chosen macOS upgrade installer using (only) Safari.

Catalina requires all apps be 64-bit. Use Go64 to find existing 32-bit apps, and then update, or replace, or retire the 32-bit apps. Microsoft Office 2011 and earlier, for instance, will not work on Catalina and later, being that they are 32-bit apps.

Alfred S Deerfield wrote:

Mr. H- Many thanks for the input. I'll read your reply a couple times as there seems to be a lot of info there! That said, it is fine for Apple to "prefer" that everyone use their newest and best, but when they obsolete other software, drivers, or peripherals which takes down a recording system, I just can't do it Apple's way.

I understand enterprise computing and production environments. That means IT (you, in this case) gets to accept the responsibility of not upgrading pretty much anything, of caching installer kits and app installers and the rest, of caching spare parts, and then dealing with the issues that increasingly arise as most computers just aren't and just can't be as isolated anymore; apps are increasingly networked, and network security means connection upgrades, and connection upgrades mean system upgrades. I'd like to still live in the buy-it-and-run-for-a-decade era, but... we don't. We're headed for software as a service everywhere, and hardware as a service for an increasing number of applications.