This is an interesting question because I have always thought that Katniss started off as a very mature young woman, having to look after and feed her family. This is of course what gives her an advantage in the Hunger Games when she goes there. However, if you are after further examples, Katniss definitely learns more about the complexities of being a contestant on the Hunger Games and how you have to play a part. For her, this means pretending to love Peeta when she is unsure of her own feelings for him. Note what she says as she is with Peeta in the cave, trying to keep him alive:
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.
This shows the awareness of Katniss of how she has to play to the audience to win, not just win.
Secondly, you might want to comment on how Katniss kills Cato at the end of the Hunger Games. We are told that it was with "pity" and not "vengeance" that she kills him, and note how she responds to having won the Hunger Games:
"Hurray for us," I get out, but there's no joy of victory in my voice.
Her sadness and grief at what has happened makes Katniss realise that being triumphant in the Hunger Games is not something to be ecstatic about.
These are two definite examples that amply demonstrate the maturity and growth of Katniss as an individual during the course of the novel.