Ponyboy is the protagonist in the novel The Outsiders, as this novel is his "coming of age" story. At the beginning of the novel Ponyboy is very childish. He argues with his brothers, runs away and reacts to Darry in an immature fashion. After the run in with Dally at the movies where Pony first meets Cherry Valance and when he stands up to Dally we begin to see some changes in Ponyboy. Later that night, after he and Johnny fall asleep in the vacant lot Pony fights with Darry. Darry hits him and Pony runs back to the lot. This is when the first climax happens. Bob and the Soc's return to the lot and nearly kill Ponyboy. Johnny kills Bob. Afterward Johnny and Pony go to Dally and eventually end up at the church in Windrixville. This is a change for Pony, as he has to deal with his actions and Johnny's actions on his own and cannot depend on Soda or Darry for help. (Not that he would have gone to Darry anyway.) The biggest changes in Pony's character happen at the church. Pony recites the Frost poem "Nothing Gold Can Stay." The symbolism here is the loss of youth and innocence. When the fire happens in the church and Pony and Johnny run back in they have shown that they no longer only care about themselves and the transformation in Ponyboy's character is nearly complete. The final step in Pony's growth into manhood is when Darry finds Pony at the hospital and they hug and have a new found respect for each other.
If I had asked the question your teacher posed this is the type of information I would be looking for. Again, this is a coming of age novel. In some ways it is the traditional boy's journey into manhood story much like Lord of the Flies is. If your teacher has been talking about stages more specifically I would suggest that you refer to your lecture notes as your teacher may have given you specific developmental stages of the character to look at.
Best of luck on your essay! This is one of my favorite novels to teach. If you enjoyed it you should continue to read Hinton's works. Rumble Fish and That was Then, This is Now both focus on the same town and same characters.