The relationship with Jack Frost is mu favorite, because of the dynamics. Bunnymund is always collected and Zen as they say, but never when it comes to the winter spirit.
Their relationships dates back and one milestone in it (or maybe it's a beginning, I don't know) is the Easter Sunday of '68, I presume we are talking here about 1968. For that blizzard, at least for most of the movie Bunnymund has not forgave him.
What it's interesting in this relationship is how they bicker, while Aster is angry with Jack, and thinks that he is no Guardian material, on the other hand he actually worries for him. An example is when the group takes the sleigh so they can reach Toothiana's palace, and Frost makes himself blow away, and even if Bunnymund is scared, he panics and tries to look where has Jack fallen, just to see him there.
Jack: oooh, you do care
Bunnymund: Oh, rack off, ya bloody show pony!
Jack is the reason why Bunnymund loosens a little (ok quote a lot), and gets again in touch with the children, something he hasn't done in too long. Jack is the one that helps him connect with Sophie, when she is found in the Warren.
One of the most heartbreaking moments in their relationship is when it's shown that Aster has begun trusting Jack as one of the Guardians, and when Frost is late for the Easter, and Pitch has destroyed all the googies, eggs that should have been hidden for the kids, Bunnymund takes it on Jack.
Bunnymund: He has to go
Bunnymund: We should NEVER have trusted you!*prepares to punch Jack, but he doesn't in the end* Easter is new beginnings. New Life. Easter's about hope. and now it's gone...
Bunnymund is hurting here, and he takes it on the only person he can, and that is Jack, because Jack is trouble, that's how Aster always saw him, and he goes again on seeing him like that.
But Jack Frost again shows that he isn't just fun and tricks, but that the actually cares for all of them, especially the Easter Bunny. It is Jack who shows Jamie that he should keep believing in Easter bunny, because he exists. And I find the moment when Jamie confirms to Bunnymund, that it was Jack who made him keep believing in the Easter bunny, one of most touching moments of the movie (and not because Aster is a cute fluffy small bunny), but how Bunnymund turns to Jack and has teary eyes, and makes that small grateful smile.
Bunnymund: He made you believe?... in me? *turning then to Jack and smiling*
After that Aster accepts Jack as part of the Guardians and as a friend.