My best quote that I can think of that would show the verdict changing their lives comes from Chapter 22. In that chapter, Jem, Scout and Dill have been called over by Miss Maudie to have cake. It is the day after the verdict and the three of them are still feeling pretty down.
While they are eating, Miss Maudie tries to comfort them. Jem does not seem to take what she says seriously. At one point he says this:
"It's like bein' a caterpillar in a cocoon, that's what it is," he said. "Like somethin' asleep wrapped up in a warm place. I always thought Maycomb folks were the best folks in the world, least that's what they seemed like."
He doesn't come right out and say it, but the implication is that he no longer thinks this. That's a major change in his life.
Another example of Jem's dissatisfaction with the jury's verdict in the Tom Robinson trial comes at the beginning of Chapter 22.
It was Jem's turn to cry...
"It ain't right, Atticus," said Jem.
"No son, it's not right."
Having watched the entire trial, Jem had the chance to make up his own mind based on the testimony given. To him, it was a no-brainer: Tom was innocent. Later, he asks Atticus, "How could they do it, how could they?" Atticus responds that it won't be the last time--
"... they'll do it again and when they do it--seems that only children weep."
In Chapter 23, after Atticus has been threatened by Bob Ewell, Jem learns that one of the Cunninghams has been the holdout on the jury.
"Golly Moses," Jem said reverently. "One minute they're tryin' to kill him and the next they're tryin' to turn him loose... I'll never understand those folks as long as I live."