The first noticeable change in the relationship between Jem and Scout is at the beginning of Chapter 2 when Scout attends school for the first time. Prior to this, Jem and Scout were often together when Jem was home. But on the way to school, Jem explained that since he was in fifth grade and she was in first, he did not want her to hang around him at recess or embarrass him in any way. This was Jem just protecting his school image from that of his private life at home but he also wanted to show Scout that fifth graders did not hang out with first graders. This kind of hierarchy is interesting because it is similar to how the children learn of Maycomb's class system in which the closed-minded population of Maycomb believed in setting social boundaries between white and black, and between poor and rich.
In Chapter 12 (the beginning of Part Two), Scout notes that Jem has been difficult. This is following Mrs. Dubose's death. Jem is twelve at this point and he's becoming an adolescent/young man. Scout narrates:
Overnight, it seemed, Jem had acquired an alien set of values and was trying to impose them on me: several times he went so far as to tell me what to do. After one altercation when Jem hollered, “It’s time you started bein‘ a girl and acting right!” I burst into tears and fled to Calpurnia.
Calpurnia explains to Scout that Jem is simply growing up. He's becoming more interested in how a boy/man is supposed to behave. And since he's getting older, he is starting to think more like an older brother than a playmate to Scout. Many of the changes in the relationship between Jem and Scout have to do with Jem's development and maturation. Their relationship changes again (Chapter 26) when Jem starts going to the high school, seventh grade, and Scout sees him less and less.