First thing: make sure the 2 versions of After Effects match! You'll have far fewer problems. In fact, if the two versions don't match, it may foil your plan. If they DO match, you'd be wise to take advantage of AE's Increment & Save feature. Save in AE the way people vote in Chicago: early & often.
You can use a thumb drive to transport the project File & media files with no problem, but don't work off the thumb drives. To make sure you're covered, you might think about having TWO thumb drives.
You could also work off a Dropbox folder. The most important thing is to keep the directory structure exactly the same. In other words if you put your working folder on the desktop ar school and have it on the desktop at home the only thing you will have to carry back and forth is the AE project. If you are working off a portable drive then you can use the collect files in AE to build a folder with all of the assets and the project file that you can carry to any computer. The collect files feature will automatically create a folder that contains your project file and all the footage. You can just move that whole folder to any computer and everything should work.
You will also need to make sure that any third-party plug-ins that you use are installed on all computers.