Well, gnomes felt they were indebted to dwarves for dwarves received them in their city when they blew up theirs.
Dwarves and humans knew eachother for a long time. They became good friends mostly due to wars against trolls and orcs that cemented their friendship.
Night elves, at the end of Warcraft 3 told everyone to gtfo of their forests. The Alliance left, as visible that even now there are few of other races there, in Classic there were only 2 humans of other races and even those were there after they had learned some customs. But since the Alliance left and the Horde did not, the night elves made friends with the Alliance. Since the Alliance did help them somewhat and didn't ask for their sacred forests or something, they tried to make the alliance work.
Draenei landed and made friends with the night elves because they had done much of the same, lived in fear of the Legion and prepared for the day they would battle them. Plus the whole betrayal by your leaders, night elves and draenei had lots in common and draenei took care to respect what the night elves believed in (like trying to clean-up the wreckage and helping them battle in Ashenvale).
Worgen, night elves felt somewhat at fault for the worgen issue. Then while many of the other races still see them with suspicion, it is because Gilneas left the Old Alliance when they were needed. But it's not enough of a reason to create great crevices. Still, Varian and Genn did fight it out.