MacOS makes this very easy. Use Time Machine and make a backup of your old Mac Mini to an external harddrive. When you setup the new Mac Mini, you will be asked if you are wanting to transfer data from an old Mac. The new Mini will transfer everything from the Time Machine backup and you will have a new Mini that is exactly like your old one (documents, programs will all be there). If you do not have a spare external hard drive, you can use Migration Assistant to copy between the computers while they are on the same network. But I would highly recommend using Time Machine.
You can also setup the new Mac Mini by itself (be sure to use the same Apple account) and then start up Migration Assistant and use the Time Machine backup to transfer your files. Instead of doing it at initial set up.
Here is Apples documentation on Migration Assistant.