In chapter 1, you should get a quest from Charles (he is a guy outside of the mage guild). No matter who you side with, they recommend you and in chapter 2 you get a letter from the academy (you might have to go to sleep before receiving the letter). After you read the letter, just go to the academy (it's in Tarkas, in the noble district) and talk to the girl at the desk. You have to pay to take the exams, so maybe save before if you try to get in.