Faye and Spike
These two look out for each other. It's debatable about why they look out for each other — do they love each other? Or do they have a respect for each other like siblings would? At the beginning of the series, Spike constantly handcuffed Faye, as he didn't trust her. He had good reason. She stole money, ate all the food, and free-loaded. It took time to earn his trust, but towards the end of the series, though it was subtle, the two cared about each other very deeply.

Faye and Jet
Jet, being the gruff and dry man that he is, does not accept Faye at the beginning of the series, either. He creates the facade of not caring about what happens to her, and makes excuses as to why he's chasing after her to bail her out of trouble. Like Spike, his respect had to be earned, and also like Spike, he takes care of and looks out for Faye, as he does for the rest of the crew on the Bebop.

Faye and Ed
Ed's presence on the Bebop was thanks to Faye, who tried to trick her to help them. Fate had it that Ed would outsmart Faye and stay on the ship anyway. Faye seems to get along fairly well with Ed, despite her tricks to try to use Ed. In a later episode, the two team up and go on an adventure of their own — Faye to discover herself, and Ed to.. well.. be Ed. :)

Faye and Ein
Faye torments that poor dog too much. She eats his dog food, hand cuffs him to ladders to keep him away from her, and generally mistreats him. Perhaps she just doesn't like dogs.Faye and Vicious
Plain and simple, she fears him. He uses her in episode 5 to lure Spike when she foolishly tries to capture him for bounty.

Faye and Julia
I believe that there is a pang of jealousy in this brief relationship. Faye and Julia team up in the second last(?) episode of Bebop, and Faye's forlorned look when she watches Julia drive away betrays her [Faye's] cool exterior.