You become an exchange student by selecting a program and applying. The "big three" of high school exchange are generally AFS, YFU and Rotary (although lately AYUSA seems to be getting a better reputation and getting closer to those three). I personally chose AFS, but I don't think you can go wrong with any of the three organizations I mentioned -- each has a slightly different application process and emphasis, but each generally has a good reputation.
When you are choosing which organization to apply to, be sure to ask for references of a student who has gone to the country you're interested in within the last year. Then ask that student detailed questions about the level of support provided abroad -- were there orientations, what services or support did the organization provide to students, did you know who to contact in case of problems, did you have any problems and how did the organization respond, etc. Support in-country is so important. When I was an exchange student, I had a few minor problems with my host school, and my organization was able to help me negotiate a solution to the problems.
Programs can be just for the summer, for a semester, or for an entire school year. You live with a host family overseas (and no, your parents don't go with you). I went on exchange for a semester, but honestly, I regret not going for an entire year. My semester as an exchange student was one of the best experiences of my life, and I do wish I could have been there for an entire year.
I do recommend going in high school if possible. You can do another exchange program in college if you'd like, but going in high school has several advantages. First, full immersion in a foreign language is better at younger ages. Second, going abroad in high school gives you more independence and a broader worldview that many of your classmates don't have. (And, if you want a practical tie-in, college admissions officers generally look favorably on applicants who've done exchanges in high school.) And finally, if you put off going on exchange, you might never go. There are all sorts of things that could happen in the next 6-7 years that could derail you going abroad in college, so why not go now? I doubt you'll regret it.