Like several other people who have responded, I think relying on OpenOffice.org Write will help you deal with your situation better than any other solution. Note that this is a full office suite, not just a word processor -- you may not need to create or edit spreadsheets, presentations or drawings, but those applications are also available.
Check out this page to understand more about OpenOffice: http://why.openoffice.org/
OOo Write will open a document created in Microsoft Word on your school computer so you can add to or improve it; and you can of course save the document in Word's .doc format.
The Microsoft Works Word Processor that you have possesses the same capability of opening and editing a Word .doc file and saving in .doc format. Well, in general, it can; there are some features in Word that Works doesn't know what to do with, so part of the document might be missing when you open it in Works.
Note too that Microsoft Word on the school's computer can open a document stored in either the OpenOffice.org or the Works native formats. To do that, you don't double-click on the file to open it, but instead use the File menu's Open command and select the right format in the dialog's File of Type list.
See http://www.microsoft.com/products/works/more/worktogether.mspx for more information.
One last note -- WordPad is yet another word processor from Microsoft that comes with your computer. It's a stripped-down editor with minimal features and a limit on the length of files it can work with. Moving files between WordPad and Word can also be managed simply (especially if you're just talking about straight text). Wikipedia describes this pretty well: http://en.wikipedia.org/wiki/WordPad