I have Visual Studio setup with a VSTO word project, and when I hit run it runs fine.
But, how do I setup a word document to use my code without having Visual Studio running?
You can do it using a Windows Installer project; starting point here. You can also use the Publish Wizard.
A more general starting point can be found in MSDN here.
You must deploy the VSTO runtime whether by hand or with an installer. Check out this blog post for a detailed explanation.