This depends on the chapter from which you are reading.
In chapter 2, when Jem takes Scout to school, we find that Jem earned some money for doing so and we later learn that on these trips to school, once in a while Jem was entrusted with his father's pocket watch.
The most significant rite of passage would occur in chapter 12 when Calpurnia begins referring to him as Mister Jem. This signifies that he has earned her respect as a man. It is at that point that he had just turned 12.
Near the end of the book, in chapter 25, Jem demonstrates maturity and gets into one of his "moods" when Scout wants to kill a roly-poly. He won't let her.
In those late 20s chapters, Jem also gets a hair under his armpit and tries to show it to Scout. She can't really see it, but pretends to be impressed.