This page lists programs that allow students to go study at other institutions during some part of their graduate career.
Exchange Scholar Program. Allows Stanford graduate students to spend up to an academic year at universities such as: University of California at Berkeley, Brown University, University of Chicago, Columbia University, Cornell University, Harvard University, Massachusetts Institute of Technology, Princeton University, University of Pennsylvania and Yale University. (Students at these schools can also use this program to spend up to an academic year at Stanford.)
Fulbright Scholarships are designed for research and language study in selected countries outside the US. They provide a maintenance allowance, as well as travel, research, and language study support. The selection process involves several stages; the first stage is done at Stanford. See the IIE website for more details and/or contact the Overseas Resource Center at Stanford's Bechtel International Center.