You absolutely have to watch Clannad, you will love it! *make sure to prepare a reserve of tissues: I cried a whole week with it. xD* Oh, and you should also watch AnoHana (Ano hi mita hana no namae o bokutachi wa mada shiranai). It's also really awesome!
And by the way, I never played any Final Fantasy. xD (Yeah, I know it's really bad! >u<)