Hey everestgal,
As someone who has a fairly large music collection I can understand your desire to save space on your computer's internal hard drive by moving your iTunes media files to an external hard drive. Let's see if we can get this set up for you.
One thing I should mention here is the possibility of using Apple's new music service, called Apple Music. This allows you to keep all of your music available online, rather than keeping it all on your hard drive. This page gives more information about the service - About Apple Music.
I would recommend, if you decide to move your iTunes media to an external hard drive, that the external drive be formatted as Mac OS Extended (Journaled.) This will ensure maximum compatibility with your computer system. You can check the drive's format by connectng it to your Mac, selecting it, either on your desktop or in a Finder window, and use the keyboard shortcut command - I or File menu Get Info. Under the General information, look for Format. If it reads Mac OS Extended (Journaled) you can skip this next step.
Note - this step will permanently erase any data on the external hard drive. If you have anything important there back it up somewhere else.
Use the steps in this article to erase the external hard drive and format it Mac OS Extended (Journaled) - Disk Utility 12.x: Erase a disk, CD, or DVD See the section titled Erase a disk.
To move your iTunes media to an external drive, use the steps in this article - Back up your iTunes library by copying it to an external drive.
Once the iTunes media has been copied to your external hard drive - and before you delete it from your internal hard drive - you need to tell iTunes where to look for the new media folder. In iTunes look in the iTunes menu > Preferences > Advanced. At the top you will see an item iTunes Media folder location. Click the Change button and in the resulting dialog select the new folder on your external hard drive. See this article for information on iTunes Advanced preferences - iTunes 12 for Mac: iTunes Advanced preferences
Once you have moved the library, test iTunes to be certain that all your media is working in the new location. Once this has been done you can move the old iTunes media folder from your internal hard drive to the Trash and empty the Trash, freeing up all that hard drive space.
iTunes will - as long as the external hard drive is connected - act just like it did before. You can play your music on your computer and sync it with your various devices such as iPhones and iPods as if the music files are on your internal hard drive. See this article about how to sync your devices - Sync your iPhone, iPad, and iPod with iTunes using USB.
Thanks for using Apple Support Communities.
Happy computing.