External harddrives are cheap these days, and even if you have no need for one now, you'll always be able to (and should) use it as a backup drive after the initial migration.
After purchasing the harddrive, plug it into your Mac or PC. Open it in a Finder or Window Explorer window, and start dragging your files over to it!
There are a few places on each platform you should look for your files.
After a cup of coffee (the more files you have to transfer, obviously, the longer it will take), you're ready to move your files to your Linux computer. (Or, alternately, you're ready to install Linux on the computer you just grabbed your data from.)
Take the media to your Linux computer and plug the drive in. You've already learned in the previous lesson how to open external media on the Gnome desktop, so open it in Files (aka, Nautilus) and start moving your data into the appropriate folders on your Linux harddrive.
It's a good idea to take the music from your Music or iTunes Media folder on the external drive, and dump that into the Music folder of your new Linux system. Take the folders and files out of Pictures and place them into Pictures. And so on. If you're migrating from a Mac, the layout will look familiar, so just stay with that convention. If you're on Windows, you'll have a better level of separation and organization, so use that to your advantage and put video files into the Video folder, random downloaded files into Downloads, and so on.