How do I go about finding satire in "The Adventures of Tom Sawyer"?
Ask yourself: Who looks stupid here?
Tom Sawyer is an allegory of the American soul. Hannibal, Missouri is about halfway up the Mississippi river, which has, for most of the nation's history. been the dividing line between east and west. As a slave state across the river from a free state, it's also the dividing line between pre-Civil War north and south.
Tom's a good kid, not like that wild, unsupervised Huckleberry Finn, but not a jerk about it like his pious, churchgoing snitch of a brother, Sid. While a lot of the characters have their directions already set for them, Tom has to figure out his own way. He's the perfect "reader identificatiom character."
One feature of allegorical stories is, every character represents some greater section of society than himself. Aunt Polly is respectable middle-class society (She owns a slave, though, so her virtue obviously has some blind spots). Injun Joe is a killer and a social outcast. Muff Potter is a drunken simpleton. Judge Thatcher represents law and order. The kids who paint the fence for Tom are the idiot unwashed masses.
Decide for yourself: Who looks stupid or bad? What part of American society do they represent? What is Twain trying to say about society with them? And that's your satire.