Connect your external hard drive to your mini
Open Disk Utility (it's in your Applications/Utilities folder)
Choose your hard drive from the list on the left. It's the line that shows the manufacturer's info for your hard drive.
Click the Partition tab.
Click the Options button, and click GUID for partition scheme.
Click OK.
Make sure Format is Mac OS Extended (journaled), and name your partition, if you like.
Click the Apply button.
That will take just a few seconds, then quit Disk Utility
And - your hard drive is completely ready for use.
And drag any files that you want to store on the external to that drive, which will have an icon on the desktop. You can organize those files by making folders for the files that you copy to your external.
Once the copy is complete, remember that the original file is still remains on your internal drive, so you can drag those on your internal drive to the trash, then, empty the trash. Even in the trash, files continue to take up space on your hard drive until you actually empty the trash.
And - you could set up your Garageband, so new files are always saved to your external drive, or just use a Save As... and browse to your external drive, so the file is saved there, and not on your internal at all.