Lalah Sune - Whether Lalah was Amuro's first love or not, she certainly taught him everything he now knows about life. Their encounter was too brief, and too early for him while too late for her. Amuro regrets the accidental murder of Lalah his entire life, although he carries that Newtype connection with her his entire life too.
Sayla Mass - Sayla respects Amuro for his abilities as the Gundam pilot, and perhaps a bit else. They pilot all the dual-pilot Gundam components together, and become close on many levels. Its my personal opinion that they fall for each other by the end, but opinions can't be proven. After the final battle, Char notices that Amuro is calling for Sayla and bids her go to him.
Fraw Bow - Had the war not occured, these two would definitely have been sweethearts. Fraw takes care of Amuro like the mother he almost doesn't having, brining him food and clothing. They're extremely close friends at first, but when Amuro goes all emotional, Fraw grows distant and hardly talks to him at all.
Bright Noa - Bright does most of his maturing in relations with Amuro. At first its a slap in the face for being cocky, then its putting the weight of every battle in the hands of a teenager. The two respect each other, but I would say there's little less than friendship here.
Mirai Yashima - Mirai watched over Amuro like a mother, or at lest like an older sister. She cares for him and tries to understand him, which is often what causes Bright's decisions to be logical. Amuro has respect for Mirai and looks up to her for recognizing him.
Kai Shiden - Kai's shouts of "Newtype! Newtype!" are less than friendly, let's just say that.
Ryu Hose - Ryu takes on almost a mentorship of Amuro when White Base leaves Side Seven, feeling that Amuro has more potential than others give him credit for. When Ryu die in, Amuro blames himself for not being a better pilot. That's a good friendship for you.
Hyato Kobayashi - (I can spell Kobyashi without help now; wow) Hyato definitely dislikes Amuro, more out of jealousy and spite than anything else. Amuro respects Hyato for the person and pilot that he his, but when Hyato and Fraw show signs of getting together, Amuro is less than pleased.