Fifteen chapters of Huckleberry Finn is a lot of novel and many "conflicts" so I will only add a few to supplement the excellent answers already provided.
One important conflict is symbolized by the contrast between the Widow Douglass and her sister, Miss Watson, who represent two types of Christians. The widow is a sort of idealized Christian who does good works and is full of Love: In many ways she represents the better side of our current philosophies, the ones which emphasize Jesus's moral and loving message and downplay the Hell-fire and damnation side of the equation. Miss Watson is the opposite. She represents a sort of holier-than-thou Christian (or any religious person) who believes that he or she is living in line with the rules of proper Christian behavior and that everyone else who is not doing the same is a sinner in God's eyes and is going to suffer eternal punishment in Hell after this life.
Sometimes the widow would take me one side and talk about Providence in a way to make a body's mouth water; but maybe next day Miss Watson would take hold and knock it all down again. I judged I could see that there was two Providences, and a poor chap would stand considerable show with the widow's Providence, but if Miss Watson's got him there warn't no help for him any more. (Chapter 3)
A very revealing conflict of sorts is revealed in the very next sentence where Huck concludes that he would be willing to "belong" to the widow's Providence "if he [God] wanted me" even though he can't understand why God would want an "ignorant" as well as "low-down and ornery" fellow like himself. This conflict is perhaps a form of man vs. himself, for Huck, who is with the possible exception of Jim about the most moral person in the book, thinks of himself the way society sees his family, as white trash. This belief that other people, people who are better educated, or go to school, or church, or hold jobs etc. are better people, and that God approves of such people and disapproves of people like Huck, well, that idea is at the base of the largest conflicts in the book. After all, we must remind ourselves, that all the so called good, church-going, proper, moral, upright people in the novel, for the most part, either own slaves or approve of the institution of slavery. For example, late in the novel, Tom Sawyer offers to help Huck in his effort to free Jim. Well, Huck simply cannot understand this: To Huck, Tom is a good boy who should not be involved in such a seriously evil (in Huck's perception, which has been warped by his society) act as helping a slave escape:
Well, one thing was dead sure; and that was, that Tom Sawyer was in earnest and was actuly going to help steal that nigger out of slavery. that was the thing that was too many for me. Here was a boy that was respectable, and well brung up, and had a character to lose; and folks at home that had characters; and he was bright and not leather-headed; and knowing and not ignorant; and not mean, but kind; and yet here he was, without any more pride, or rightness, or feeling, than to stoop to this business, and make himself a shame, and his family a shame, before everybody. I couldn't understand it, no way at all. It was outrageous... (Chapter 34)
This passage reveals so much of what makes Huckleberry Finn the great American classic that it is. Huck's sincerity and genuine good feeling, indeed love, in wishing to free Jim from slavery is the most moral and upright thing in the novel--yet he sees himself, and his plan to free Jim, as the exact opposite of that, as dirty, low, immoral, shameful, literally disgusting! This is Twain's ingenious indictment of the Southern Society he was raised in--that it taught everyone who wanted to live in that society as an accepted member of the social world that slavery was good, and that those who opposed slavery or aided slaves in escaping, were evil--when Twain could see and wanted his readers to see that the exact opposite was true: That slavery was evil and that those who opposed it or aided slaves in escaping were good! This conflict, that Southern Society was teaching that what we know to be evil (slavery) was good, and what we know to be good (undermining slavery) was evil is the true heart of this amazing novel. It creates the single greatest conflict in the novel, Huck versus himself, because Huck's natural goodness knows that the right thing to do is to set Jim free, but his conscience, which would normally tell a person what is right or wrong, has been warped by his Southern upbringing so horribly that he feels guilty for helping Jim! Thus, in the greatest chapter in the novel, Chapter 31, when Huck decides he will literally go to Hell if that is the price he must pay for helping his friend, we should remind ourselves that Huck has been raised among church-going, strict Baptists for whom Hell is a real and terrible place--eternal damnation--eternal suffering and torture--a place where one burns forever! How tragic, that a good boy, doing the right thing, should be made to feel that he is evil and degenerate for actions that are objectively moral!