At first, I dreamed a fantasy of foreign countries only. However, later I recognized that the exchange student program was a tremendous positive opportunity to me.
At first, I was fascinated by the new experiences in foreign countries. However, later I realized it offers much more positive opportunities to an exchange student;Since you are going to discuss about those positives in forth coming paras,have the semicolon.
First, the program can give me a chance to study about my major courses with foreigners.
... you better say what the benefit of studying them with foreigners. That's the most important point here. Also, since you use the word "first" in the previous sentence, it sounds too repetitive when you use it again. This is my suggestion;
I'm looking forward to study my major courses with foreigners and this would benefit me .... ... tell how it's going to benefit you
In addition, I believed that competition and cooperation with foreign student is a special and rare experience. Also, I knew that the exchange program is a just one offer in my entire life: because I could not be a university student twice.
Further, being in competition and cooperation among foreign students is a rare experience that you would not get for a second time in life.