I have a Mac mini - with High Sierra 10.13.6 I can't seem to get any updates - or is this it??

What model year is it?

The model year is available at Apple menu > About this Mac.

A 2010 and 2011 Mac mini cannot upgrade beyond High Sierra.

Models later than 2011 can upgrade to later systems, depending on which model year it is.

See the following for the latest compatible operating system for each model year: Identify your Mac mini model - Apple Support

Apple stopped issuing updates for High Sierra several years ago.

As for upgrades, I suspect a computer of that vintage may have reached the end of the line. Use the computer information, including the model year, and the links below to determine what is possible with your computer.

Click on this link to read how to get different versions of macOS --> How to download macOS - Apple Support

- The article also has links to system requirements for each OS version which you should check against your exact computer model. For any model produced in the last 5 years or so you should use the system requirements links for each OS version instead. This web site also has a reliable presentation of model and OS version compatibility: https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

- It helps to have more RAM (memory) than what is noted as "minimum" for any particular system or your computer may run slowly.

- You may find you must use Safari to get links for High Sierra and later using the Mac App Store application to open to a functioning page.

- Sierra, El Capitan, and Yosemite download as a .pkg. Click on this to have a converted version placed in your Applications folder as an "Install OS..." application. Run the installer and follow the directions.

- Upgrades may take some time (up to an hour or more) to install, with restarts and blank screens.

Before upgrading, back up at least your data in case something goes wrong. See this support document: How to back up your Mac - How to back up your Mac - Official Apple Support

Check if your old apps will still work with with the OS you want to install - https://roaringapps.com/apps This is particularly critical if you are upgrading to Catalina or newer, as older 32 bit apps will no longer work. To check for those, try this 32 bit application scanner application --> https://www.stclairsoft.com/Go64/

If an error occurred while updating or installing macOS --> If an error occurred while updating or installing macOS - Apple Support

You could try installing a Linux distro on it (Probably not supported by Apple any longer). I just installed the latest Debian OS 12.2 on a 2008 Mac Mini (3Gb RAM) and it runs pretty well - it's running an old 5400RPM hdd too. Installing an SSD will probably help a bit on performance. Check it out if you have an old Mac laying around that you want to bring new life to.

https://www.debian.org/

The latest versions of Chrome and Firefox (Even Microsoft Edge!) can be installed on Debian.