For the purpose of answering this question, I will assume you are referring to the novel The Hunger Games rather than the whole trilogy.
The first event that leads to Katniss's character development is when her sister's name is pulled out of the bowl at the annual reaping, meaning that Prim is to be District 12's female tribute in that year's Hunger Games. Katniss volunteers to take her sister's place, showing that she is willing to die to save her sister.
The next event that I would say shows Katniss's character development is the first time she decides to take action to kill. She is hiding up a tree in the arena when a fellow tribute, Rue, points out that there is a nest of tracker jackers further up the tree that she was in. By sawing off a branch and sending the nest of these deadly creatures to the ground, where her pursuers are sleeping, she shows that she has accepted the need to kill her fellow tributes.
Katniss shows character development once again toward the end of the Games, when a new rule is introduced that will allow her and her fellow District 12 tribute, Peeta, to both return home alive. Katniss, who has been very much a lone wolf up until this point, seeks out her fellow tribute, risks her life to nurse him back to health, and begins to open her heart to him.