Hello, rehouse!
Welcome to Apple Support Communities, and congrats on the new Mac Mini! From what I'm understanding from your post, you have plugged your iPod shuffle into your new Mac Mini using its USB cable, and you are not seeing it show up in iTunes. You can't sync songs to your Shuffle without seeing it in iTunes, so I can see why you're concerned! I would be happy to help you get your iPod Shuffle synced and working with a few suggestions.
First, make sure you have the latest version of iTunes on your new Mac Mini: Get the latest version of iTunes
If it's still not showing up after that, this document has some suggestions: If iTunes doesn‘t recognize your iPhone, iPad, or iPod
This page has tips specifically for the Shuffle: iPod shuffle Support
The Mini has 4 USB3 ports, so make sure to try a different one if you haven't already. If you have another Apple device (like an iPhone or iPad), try plugging it in to see if iTunes on your Mini recognizes it. That will help you know if it's only the Shuffle that's not being recognized, or something going on with the Mac Mini or its ports.
Have a great week!