In The Hunger Games how does Katniss develop from the beginning to the end of the book?
Considering The Hunger Games, the first book of the trilogy, alone, it is in fact difficult to see Katniss's development. However, she is not the same person at the end of the Mockingjay, and in retrospect, the reader can see that the changes began during the events described in the first book.
As mentioned in the previous answers, Katniss is not afraid to break rules and she is single-mindedly focused on her and her family's survival. Even her volunteering to take Prim's place in the Games might have been more due to practicality, rather than sentimentality—Katniss is older and more likely to survive. This is not surprising given that survival has always been a struggle, and there was simply no time, or strength, for anything else.
This starts to change during the Games, when the struggle is not simply to survive, but to survive without becoming a murderer. At first, when Peeta mentions to Katniss the night before the Games that he does not want the Games to change him—to make him a savage—she is annoyed. But as she meets other tributes, particularly Rue, she begins to understand and relate to what he meant. So, her world is no longer black and white. She develops connections to people outside of her family. She realizes that Peeta has known her for years—she has been "a subject of dinner conversations" in his family—and her actions had left imprint on the residents of District 12, something she never considered before.
As the series progress, Katniss becomes less and less judgmental and narrow-minded. She becomes much more forgiving of other people's faults, such as Haymitch's drinking and the silliness of her "prep team." She begins to try to forgive her mother, whom she resented since she was too depressed to care for her children after the death of Katniss's father. While in District 13, she even develops friendship with Johanna Mason, a tribute from District 7, who she originally despised. In other words, as the story evolves, Katniss becomes compassionate and perceptive. This perception might have been what enabled her to see Alma Coin's impeding takeover of Panem for what it was, and change the course of history single-handedly by killing her.
I must agree with deem1510 in that Katniss does not sprint across the finish line but limps through it. At the end of the trilogy, having lived through the violence and the constant danger of the Games and the war, she is suffering from post-traumatic stress. But she still finds the courage to face each day, to help rebuild the District 12, and to raise children, despite her great reluctance to take this risk.