Must I make up for the day I broke my fast in Ramadan because I was unaware of the beginning of the fast in America, where I study, knowing that my home country had not yet started fasting?

Your obligation is to fast as soon as the crescent moon of Ramadan (definition) is sighted in your country of residence. Therefore, you were correct in abstaining from food and drink as soon as you learned that fasting had been established. As for the day you broke your fast out of ignorance, you are obliged to make it up according to the majority of scholars, and this is the most preponderant opinion and closer to caution.