Two moments from the novel immediately come to mind - the Grangerford episode and Huck's escape from Pap Finn.
Early in the novel, Huck is challenged to escape from the abuse and exploitation of his father, Pap. To pull off the escape, Huck has to fake his own death. This is not merely a feat of wit and cunning, but also a moral challenge as Huck must come to terms with a complete break from his past (including his friends and guardians).
Huck's willingness to fake his death and take the risk of escape demonstrates his adventurous spirit and his intelligence. He copes with the aftermath fairly well, especially after finding Jim on the island.
This episode is important for many reasons, one of which is that Huck's desire to be free from undue oppression is articulated clearly.
Later, Huck encounters an ideal family in the Grangerfords only to discover that they are driven to madness by a feud. Outside of the feud, the family is kind, intelligent and generous. However, they choose to maintain a running dispute with another family that leads often to death and always to danger.
The significance of this episode is also multifaceted. One false idea is enough to destroy a whole family. This idea recurs in the novel in various guises.
Huck learns here also that love and hate can coexist. Kindness and rampant aggression coincide in the Grangerfords. Associating himself with the family, Huck puts himself in physical and moral danger. This danger is also repeated throughout the novel, but Huck meets it in relative innocence at this point in the novel.