How does Jem and Scout's relationship change as Jem grows up in To Kill a Mockingbird?

At first, Scout is the "little sister" Jem takes to school and advises. At recess on the first day of school, for instance, a disgruntled Scout complains to Jem about her teacher Miss Caroline. Jem tries to allay her fears, telling her not to worry because his teacher says that Miss Caroline is introducing a new method of instruction called "the Dewey Decimal System." (He has mistaken John Dewey's pedagogical method with the system of arranging books in a library.)
In Chapter 7 Jem reassures Scout as she begins second grade, telling her school gets better as she advances.
Jem is the voice of wisdom to Scout at times. For instance, when Scout attacks Walter Cunningham by rubbing his nose in the dirt for causing her to be punished by Miss Caroline, Jem stops her and invites Walter home to eat with them at noontime.
In another instance, Jem scolds Scout for taking gum from the Radleys' tree's knothole.
As the protective older brother, Jem allays Scout's fears at times. In one instance, he tells Scout that she does not need to be afraid of Boo Radley because nothing can "get her" with Calpurnia and him at home during the day and Atticus there at night. As a result, Scout feels Jem is a "born hero" (Chapter 4).
Scout tags along with Dill and Jem, who are the architects of various schemes regarding Boo Radley. She acts as a lookout while Jem goes onto the Radley porch, for example, in Chapter 5. 
Scout gets used as the "test pilot" for some of Jem and Dill's ideas. For instance, when the boys have a large tire, they send Scout down the hill inside the tire first.
As Jem begins to mature, he is at times moody, and Scout must leave him alone, per instructions from Atticus. Nevertheless, Jem confides in Scout in Chapter 7, when he tells her about his pants that he has torn and abandoned as they were caught on the Radleys' fence. He shows Scout that the pants were sewn for him by Boo. Further, he shares his maturing perspectives. For instance, whereas he earlier told Scout to throw away the chewing gum left for them in the knothole, in Chapter 7 when Scout hurls the soap carvings onto the ground, Jem scolds her, saying, "These are good," and he later stores them in his keepsake trunk at home.
Although Jem changes, he remains loyal to Scout. Feeling Scout is justified in her actions, Jem allows Scout to fight her own battle with Francis in Chapter 9 after Francis calls Atticus a pejorative term.
There is camaraderie between Scout and Jem, as they both receive rifles for Christmas, attend church with Calpurnia, and go places together.
Although he remains brotherly, Scout finds Jem moody as he enters puberty:

His maddening superiority was unbearable these days. He didn't want to do anything but read and go off by himself. Still, everything he read he passed along to me, but. . . now for my edification and instruction.

The relationship between Jem and Scout becomes more divided when he is a teen and she is not. He is moodier, and he speaks in a condescending manner to and about Scout. At Tom Robinson's trial, Jem tells Reverend Sykes that Scout does not understand some of the language, questions, and answers given by the witnesses.
After the trial, Jem's ideas begin to alter. For example, he scolds Scout for having crushed a roly-poly (Chapter 25), and she attributes his superior attitude to "part of a stage he was going through." Further, Scout is angered by Jem's reporting of Dill's presence in Scout's room after he runs away.
The loyalty and love between Jem and Scout remains intact, nevertheless.