Is there a way that I can save my Dreamweaver CC files so that I can work on them in CS6 at school later?
You could save them to a thumb drive at home, then overwrite the local copies of the school machines root folder with the modified files.
You could also Put (up arrow in Files window) to a remote server from home, then Get (down arrow in the Files window) those files from the server while at school and overwrite your school machine's local files that way and vise versa.
HTML is HTML, whether it was written by DWCS6 or DWCC, there's no "export for CC" or anything like that. The only things you'd have issues with are those items that have changed between versions. If you are using Spry for instance, CC won't know how to deal with those parts of the code, so you won't be able to make modifications using the dialogue boxes that appear in DWCS6.