It is the student's responsibility to obtain the necessary visas for foreign travel. The best place to obtain information is from web page of the country's embassy in Washington, DC as well as from the information from the organization and / or university. For a list of visa requirements and foreign embassies and diplomatic missions click here.
Passports are issued by the US Department of State. You can apply in person at a US Post Office or by mail (renewals only, certain restrictions apply). Even if you're "just going to Canada or Mexico" you need a passport or passport card. Are you ready?