What is the significance of each of their first words?

It seems as though Amir's first word indicates his affinity for and responsibility towards his father. This culminates in his concern for not fulfilling his father's understanding of what a "man" is. In Hassan's case, it seems clear that he is also denoting his responsibility--only in his case, it is responsibility towards a friend.

Amir explains that his first word was 'Baba' and Hassan's 'Amir', suggesting that Amir looks up most to Baba, while Hassan looks up to Amir.