Use one of the many study abroad directories available online to browse for a program that fits your desires. The links below are a few good places to start looking. Always check to see exactly what is included in the program costs, so you know what other expenses you'll need to be prepared for.
Once the desired program is found, assess what is required to apply. This may include a certain GPA, letters of recommendation, an essay, forms to fill out, documents to send and other things that are similar to applying for college in the United States. If the student is already attending a school that happens to offer the overseas programs/exchange that they want, then it can make the application process easier, since they already have most of the student's necessary information and documents.
Apply to the program. Once accepted and if needed, look into whether financial aid is available specifically for studying abroad. It's possible to use regular financial aid as well. Grants and scholarships are also out there for interested students.
After celebrating your acceptance and getting the finances in order, it's time to start planning your adventure. Depending on the program, room and board may be included, or you may have to find a temporary dormitory or apartment, which staff at the school abroad can help with. Field trips may also be included, or at least an option to sign up for, and are a perfect way to explore the area.