I just received my new Mac mini and I'm having trouble connecting it to my older iMac display

You can't use an iMac as a Target Display Mode monitor for a new (Apple Silicon) Mac Mini.

Even if your iMac supports Target Display Mode, there's a restriction that "The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed."

Use your iMac as a display with target display mode - Apple Support

There is a third-party product called Luna Display.

https://astropad.com/product/lunadisplay/

It's $120 (USB-C version) and is "NOT recommended for gaming, video playback, or other high-motion content." It also might tie up a couple of ports on your new Mac (one for the device, one for a connection to the old Mac).