How do I upgrade a recently purchased Mac mini to IOS 15?

The name of the operating system is unmistakable when you select the Apple () menu > About This Mac. If your HP MFP is networked and appears on the built-in Apple AirPrint list, then your driver would be named AirPrint, and there would be no need for any HP driver downloads. If this is not the case, you will need to look up your particular MFP model at the HP site, enter the version of macOS you want driver compatibility and then download the driver, and reset your printing system so the HP device can see the new drivers.

You can upgrade to macOS Monterey if your Mac mini is a Late-2014 model and as far as Big Sur if your mini was generally released in 2014. Otherwise, you will need a Late-2012 model mini or newer in order to upgrade to Catalina. See How to get old versions of macOS using strictly the Safari browser.

iOS 15 is for iPhones.

See: How to upgrade to macOS Big Sur - Apple Support

Second, if you are not able to use your printer or scanner, the solution is normally to update the devices driver/software to one that is compatible with your operating system. Check the HP support web site.

VikingOSX wrote:

You can upgrade to macOS Monterey if your Mac mini is a Late-2014 model…

Did I miss something?