Flash drives are your safest bet for data transfer, and even public school computers should have USB ports.
CD-Rs are also okay for occasional use, but their "write-once" nature makes them a bit wasteful for documents that are likely to change, as a new CD must be burnt each time. If you decided to use CD-RWs, you might find that some discs work in some computers but not in others.
The only thing you should be aware of when using Flash drives on Linux is that if you copy over a large file to the drive, then try to unmount it, it may take a while to unmount. This is because the file is still copying in the background! Don't worry, this is a feature of Linux, and your drive will unmount when copying really finishes.
I have twins entering high school. My computer is dated to say the least. What are the most popular forms of data storage; CD, DVD, floppy, zip drive, flash drive. etc. used by public schools these days?