The exposition is the part of a story where background information on the characters and situations is presented. Although the plot structure of "Adventures of Tom Sawyer" is a little different from the traditional in that it is made up in parts by a series of vignettes, I would say that the first seven chapters, in which the characters and their personalities are introduced, might serve as exposition for the main storyline.
The climax, or point at which the main conflict reaches its greatest intensity, would be when Tom and Becky disappear on their excursion to the caves in Chapter XXXIX, and the resolution, when the main conflict is resolved, would take place in the final three chapters, when Tom and Becky are found, Injun Joe's fate is revealed, and Huck is taken in by the Widow Douglass.
The rising action is the development of the plot from the exposition to the climax, or, in other words, everything that occurs between those two points. In this particular book, the rising action would include the murder at the graveyard, the trial of Muff Potter, and the boys' search for the treasure and Injun Joe. The falling action is everything that occurs between the climax and the resolution, and includes the search for Tom and Becky and Tom's "funeral".