Compassion abounds in The Fellowship of the Ring. Of course, we are reminded of Bilbo's compassion toward Gollum when Bilbo first came into possession of the Ring in The Hobbit. But Frodo's crew follows in his uncle's footsteps. Gandalf encourages Frodo to continue this compassion for Gollum, who is depicted as a victim of sorts (despite the fact that he, you know, murdered his best friend.) Galadriel, an Elf, even finds it in herself to feel compassion for Gimli because of his people's loss of the Mines of Moria. And Gimli hates Elves! Most importantly, though, Frodo's friends all feel compassion for the position Frodo is in: they know he didn't choose his situation. They, however, do have a choice; and they make the compassionate one: join their friend instead of leaving him out to dry.
In The Fellowship of the Ring, compassion is a good thing in and of itself, whether the target of that compassion appreciates it or not.
Galadriel's compassion to Gimli for his people's loss of the Mines of Moria heals Gimli of all of his hatred for Elves as a people.