This is a false cognate, if ever I've known one. Фамилия means not "family", but rather "last (family) name". The word for "family" is семья.
I think of фамилия as family name, which is usually the surname of a person
yes and so is the pronoun proceeding it. once I learned the case endings and learned to identify them Russian became much easier to read (feminine accusative: U/YU
Is there some table to understand better the cases and the relative endings? Please, going crazy over here!
I particularly liked this one, which was posted by someone else in another excercise: http://www.russianlessons.net/grammar/nouns_nominative.php
The other cases are linked on the page.