Edgar: Twin brothers. Best friends. The single closest relationship in the whole game. It's illustrated at every turn, both in the game and in this website, so I don't think I really need to elaborate much.

Cyan: Sabin is probably Cyan's only close friend within the party, as I can't see Cyan cozying up to a thief, a would-be lady's man, a couple of former Imperials, an assassin, a moogle... you get the idea. Sabin tries to be understanding of the old guy, though he can't help teasing him affectionately.

Gau: Sabin sort of adopts him as a little brother, I would say. If Sabin's not in your party, you won't get Gau's custom scene at his father's house. Of course, that scene doesn't really turn out as Sabin hoped, but he meant well.

Terra: They seem pretty friendly in the beginning of the game, but don't interact much after that. If you've explored the links, you'll have noticed that I was able to locate a couple of fansites that pair the two of them off. I think it's kind of cute, myself.

Celes: In Zozo, after you find out about Terra and the Empire's magic-extraction process, Sabin asks Celes about it. Also, they're the first two you can get in the World of Ruin. I get the impression they're friends, but they don't interact much where we can see it.

Locke: I'm guessing he teases Locke about being a thief, judging by some of his comments (mentioning that Locke would be jealous of Gau's "shiny thing," for instance) but they don't seem especially close.

Strago: No interaction to speak of.

Relm: I think he's annoyed by her.

Gogo: No one interacts with Gogo.

Umaro: Umaro only interacts with Mog... I guess.