In the novel The Hunger Games, how do the characters become more mature or responsible throughout the story?

I'd like to focus on Katniss Everdeen for this question. In some ways, Katniss does not do much maturing during this novel. This is because she has been forced into a coming-of-age moment much earlier than a typical teenage girl. Her father was killed in a mining accident when she was 11, and her mother became a shell of a person. This forced Katniss to grow up quite quickly because she became the primary caregiver to her mother and sister. Katniss did most of the work when it came to getting food, small amounts of money, or trade items for the family.
With all of that said, Katniss does indeed continue to mature during the Games. When she first gets to the Capitol and into the arena, she is counting on her own skills and wits to keep her alive. She isn't keen on playing the political game that Cinna and Haymitch stress so much. Katniss eventually learns the critical need of sponsors, and she matures enough to recognize that she has to give the audience more than just a struggle for her life. She has to give them romance as well. This is why she starts playing up that angle with Peeta.

If I want to keep Peeta alive, I've got to give the audience something more to care about. Star-crossed lovers desperate to get home together. Two hearts beating as one. Romance.

While this "romance" might have been initiated for survival reasons, Katniss begins caring for Peeta and his survival. She feels that he is her responsibility. She also feels this way about Rue. Those feelings might not have been present at the start of the games, but this change in attitude does reflect Katniss's general character trait of caring for those that she is close to. She cares for her mom and sister, and caring for Peeta and Rue is a natural progression.