Egill's relationship with his family was rather unusual. His mother praised all his doings. When he had killed his first man, only six years old, his mother said that he was an energetic boy and he would certainly become a brave viking when he'd grown up!
Egill's father didn't share these opinions of his wife and tried to raise Egill properly but then Egill got stubborn and didn't utter a word to his father for a whole year.
Egill and Thorolfur, his brother, were good friends but I think that Thorolfur was a bit afraid of Egill because Thorolfur always did what Egill wanted and never did anything against Egill's will.
Not much is told about Egill's relationship with women. I think it's because Egill was not the kind of "soft", handsome guy that women would like. But once in the saga there's a story about a party where Egill got to know a beautiful daughter of an earl and their relationship at this party seemed be to excellent.