Never work off a removable drive.
Copy everything to the harddrive.
Although I would generally endorse the advise of Bob Levine, I don't think that this is the core of your current problem.
If you move the InDesign document to a hard drive and open it in InDesign, does the problem still occur? If so, the problem has to do with where and/or how the fonts are installed.
Please provide specifics with regards to OS, where you are installing these fonts, version of InDesign, etc.
I’m using InDesign CC 2014 on a Microsoft Windows computer.
I have tried installing the fonts through the Control Panel and also directly through InDesign, but neither method worked. I did try again after transferring the files to my hard drive, but that didn’t help.
Do the fonts show up in the Windows Fonts Control Panel?
Are those fonts available to you in other applications (such as Microsoft Office applications) after you attempt to install them?
And of course you exit InDesign before installing the fonts. InDesign can't find fonts that are installed while it is running.
Thanks for the insight. I discovered that the reason the fonts weren't downloading is because the supplied fonts are Mac-based, and I"m working on a PC.