I need help finding quotes that would show Huck's gradual progression throughout the story.
In chapter 15, after Huck has played a mean trick on Jim and Jim calls him out for it, Huck says,
It was fifteen minutes before I could work myself up to go and humble myself to a nigger—but I done it, and I warn't ever sorry for it afterwards, neither.
Huck has recognized Jim's humanity. He understands that Jim sees him as a friend and that he has truly hurt Jim's feelings. Huck has to work hard to unlearn all the racial attitudes he has been surrounded with all his life.
Near the end of chapter 23, Huck witnesses Jim's grief in the early morning hours and intuits that Jim is "low and homesick" and likely thinking about his wife and children. Huck thinks,
...I do believe that he cared just as much for his people as white folks does for theirn. It don't seem natural, but I reckon it's so.
This observation is another indication that Huck sees Jim as a fellow human being and not someone lesser than himself or white people in general.
Chapter 31 holds perhaps the most dramatic turn in Huck's new way of thinking and valuing. He writes a letter to Miss Watson that discloses Jim's whereabouts, and momentarily, he feels, "light as a feather" because he has been conditioned to believe that slavery is sanctioned by God. It is what some Southern churches were teaching, and Huck believes that he is cleansing himself of sin by arranging for Jim's return to his owner. However, Huck's larger moral conscience will not allow him to send the letter, and he tears it up, saying,
All right, then, I'll go to hell...
In reflecting on the relationship he has developed with Jim, Huck promises himself to continue to work to keep Jim free. This is a monumental moment for Huck because he has been conditioned to believe that working to free Jim will damn himself for eternity.
A prime example can be found in Chapter Sixteen and reflects Huck's increasing moral maturity. He muses: "Conscience says to me 'What had poor Miss Watson done to you, that you could see her nigger go off right under your eyes and never say one single word? What did that poor old woman do to you, that you could treat her so mean?...' I got to feeling so mean and so miserable I most wished I was dead." This shows Huck's burgeoning conscience.
A second example comes from Chapter 19, when Huck comes to the understand true nature of the "king" and the "duke." He says, "These liars warn't no kings nor dukes, at all, but just low-down humbugs and frauds." The idea will help Huck mature and be more wary of such unsubstatiated claims in the future.
A third example can be found in Chapter 40. Though he still has some growing to do in terms of morals, Huck is finally beginning to discard the prejudices with which he has grown up. When Jim is hurt, Huck decides, "I knowed he was white inside, and I reckoned he'd say what he did say - so it was all right, now, and I told Tom I was agoing for a doctor."