In my opinion, Huck's opinion of slavery doesn't really change over the course of the novel. He is not, in general, opposed to slavery at the begninning and he's really not opposed to it at the end either. Throughout, though, he seems to make an exception for Jim.
When Huck meets Jim on the island in Chapter 8, he promises not to tell on him even though that will make people despise him. But he doesn't say anything about not liking slavery. And before that, in Chapter 2, he talks about how Jim was almost "ruined as a servant" because he got stuck up after he thought he was bewitched. None of this really sounds anti-slavery.
At the end of the book, Huck wants to free Jim, but he also seems to think that it's wrong. He is surprised when Tom is going to help him free Jim:
Well, I let go all holts then, like I was shot. It was the most astonishing speech I ever heard -- and I'm bound to say Tom Sawyer fell considerable in my estimation. Only I couldn't believe it. Tom Sawyer a nigger-stealer!
He wants Jim to be free, but doesn't seem to see it as an issue of slavery per se being wrong
Once I said to myself it would be a thousand times better for Jim to be a slave at home where his family was, as long as he'd got to be a slave...
You would think he wouldn't put it this way if he thought slavery were completely immoral in every case.
And he doesn't seem to have any problem with the fact that other blacks are slaves -- it's just part of life.
So I'd say his opinion doesn't change -- he's always okay with slavery, but wants Jim free, presumably because Huck is "enlightened" enough to like individual blacks and want them to be free.
Originally, Huck believes that it is his duty to turn in Jim, a runaway slave. He does not simply see it as obeying the law; he actually believes that it would be immoral for him not to turn in Jim. His morality has been shaped by a society with flawed values.
Later, after Huck matures, gets to know Jim better, and encounters various situations that shed more light on right and wrong, he realizes that he must base his actions on his conscience rather than on what he has been taught is right. A pivotal moment in the novel which illustrates Huck's developing view on slavery is when he goes to turn in Jim to the runaway slave hunters and decides at the last minute to fib and return to Jim.