In the first fifteen chapters, the author sets up many of the conflicts that will appear again and again in the novel. Perhaps the most important conflict we see is man vs. society. Huck Finn is being “sivilized” and he is not sure if he likes it or not. For example, he likes going to school and learning, but he is told by his Pap learning is worthless. “And looky here, you drop that school…” Huck enjoys being warm and dry, but he does not enjoy the rules that go along with living in the widow’s house. “I liked the old ways best, but I was getting so I liked the new ones, too, a little bit.”
The man vs. society conflict can also be seen in the game of robbers, the brain child of Tom Sawyer. The elaborate game shows a desire to go against the rules of society, yet at the end of the chapter, the boys decide it would be “wicked to do it on a Sunday.”
Another place the reader can see the man vs. society conflict is in Huck’s reaction to meeting Jim on the island. Huck is startled by Jim’s revelation that he has run off, but he promises not to turn Jim in, even though society views that decision as wrong. “People would call me a low-down Abolitionist and despise me for keeping mum, but that don’t make no difference.” Huck knows society would not approve of his decision, but he makes it anyway. In the first fifteen chapters, one can also see the internal (man vs. himself) conflict Huck is facing. Huck wants to do the “right thing,” but he is unsure if the right thing is what society tells him or what his heart tells him.
After Huck plays a trick on Jim, making him doubt himself, Jim scolds Huck. Huck knows he has done something wrong. “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 afterward.” This sets up the internal conflict for the rest of the novel: is Jim property or is he a person?
The major conflict in the first fifteen chapters is Huck's need to escape his father.
Pap Finn is an abusive drunk and he wants to get Huck's money. While Huck does enjoy the ease and freedom of living without social rules when he is with his father, he fears what his father might do to him and he is forced to work much more than he does with the widow.
During the brief time that Huck lives with his father in the cabin, Huck is nearly killed by Pap during a drunken episode.
Huck's need to escape with his life and his freedom (and to some extent with his money) represents the main conflict of the opening section of the novel.
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!