Throughout The Kite Runner, Baba worries because Amir never stands up for himself. When does this change?

While Baba worries that Amir does not stand up for himself, the true problem is that Amir cares too much about himself and does not stand up for others (i.e., Hassan).  Amir does struggle with Assef's intimidation, but he is so self-centered that he cannot bring himself to defend Hassan.

This internal conflict on Amir's part changes when he reaches the United States.  He stands up for his right to choose his major when he decides upon writing instead of medicine--a decision Baba strongly dislikes.  He also takes the initiative to choose his own bride and asks Baba for help after he has made the decision.

Of course, Amir's ultimate victory is when he stands up against Assef for Sohrab's rescue.  At this point, he attempts to redeem himself for being a coward so many years ago.