Upgrading from 10.14 to 13

Your 2014 Mac mini can only upgrade to macOS Monterey (12.6.8) as it is not supported in Ventura. There are no Apple trials as upgrading the operating system to Monterey will convert the filesystem to APFS and replace Mojave.

As Mojave was the last operating system to support 32-bit applications, you should check that any 32-bit applications or suites have 64-bit counterparts to work on Monterey. I suggest that you run the free Go64 application to get a list of those 32-bit dependencies, and also check with your printer/scanner vendor to determine if they still provide drivers for macOS 12 compatibility. Your printer, when networked, may also be supported by Apple's built-in AirPrint and if so, would replace the need for any vendor drivers.

Perform a final Time Machine backup before the upgrade, and have a replacement Time Machine drive available (formatted as APFS) to begin new Time Machine backups on Monterey. Do not reuse your Mojave Time Machine drive. I also recommend booting Mojave into Recovery mode and running Disk Utility First Aid followed by a normal reboot. Then…

You can get the full installer for Monterey here: How to download and install macOS - Apple Support

The Monterey installer should reside in your /Applications folder afterward.

ghn wrote:

I finally got word from the developers that the app is being upgraded this year so that it will be compatible with Ventura (13).

Which means that you won't be able to run the upgraded application on your current computer. The Mac mini (Late 2014) can only go as far as the latest version of macOS 12 (Monterey) – no higher.

If you want to upgrade the OS for the sake of compatibility with other new applications, you can do that. But you indicated that the application that you use "A LOT" is not compatible with Catalina, Big Sur, or Monterey. So upgrading would mean losing all access to that application, which is probably something you want to avoid.

ghn wrote:

I didn't understand what you [VikingOSX] said about "hav[ing] a replacement Time Machine drive available."

What he's saying is that you want to have two Time Machine drives available.

You want to have an up-to-date Time Machine backup from the Mojave installation on your old Mac Mini, as a source that Migration Assistant can migrate your data from. For this purpose, an up-to-date bootable backup made with Carbon Copy Cloner or SuperDuper! would work, as well. The basic idea is, Migration Assistant can retrieve your old system's data from a backup ... it doesn't need a direct connection to the old Mac.

If you're going to start doing Time Machine backups of the new system, you want to do that to a newly-erased drive (what he's calling the second Time Machine drive). You don't want to have Time Machine start mixing in backups of the new Mini that's running Ventura, with the backups of the old Mac that was running Mojave. Especially since there have been changes in what filesystems Time Machine supports for use on new backup drives (used to be HFS+; then HFS+ or APFS; now APFS).

Maybe you could mix oil and water, but he's suggesting it's best not to try. Keep your backups of the new Mini separate from your backups of the old one.

My recommendation to you (after you check 32>64bit availabilities) would be to go ahead and upgrade to the highest OS your mini will take, which is Monterey. It's still supported by Apple and gets security and native app updates Safari, Mail, etc., and will be for a couple more years.

In my opinion it's been a very good, simple, OS. Your mini would be better protected for a few more years, unlike now with Mojave. It is widely supported by 3rd parties (printer drivers, etc)

You can always get a second new machine with Ventura which, from what I read, has had a bit of a learning curve.

I'm sticking with OS12 for the duration and as long as possible.

If and Only If the said Application that is going to be updated and will only run in macOS 13 Ventura

Follow previous advise - First

Or consider purchasing a New Computer with Ventura already installed >>

Explore the Apple Trade-In  Process where the Value of the exiting device could be used against the cost of a new Apple Computer, iPhone or iPad 

Notation - Some devices and depending on the computers age may have little to no value at Trade In time.

But at least the device will not arrive to the Land Fill Site and be recycled in a responsible way by Apple

Hey, still running a 2011 Mac Mini with some Modifications like 8 GB RAM and replaced the Fusion with a 500 GB SSD.

Like yours - Zero value

Best of luck on whatever machine you purchase