1. Jem and Scout meet Dill and the children embark upon their quest to make Boo Radley "come out." They encounter some issues with his brother Nathan, who shoots at them, but later Boo starts leaving the children little gifts in the tree hollow.
2. Scout has trouble at school and during the story of her many school encounters, we are introduced to the Ewell family who will become more important later in the novel.
3. Jem and Scout receive air rifles for Christmas but are told not to shoot them. Soon after, Atticus shoots Tim Johnson, the mad dog, earning a great deal of respect in the eyes of his children. This is important because it foreshadows events in the later chapters of the novel when Atticus proves his strengths include more than just being a good shot.
4. Atticus is chosen to defend Tom Robinson who has been accused by Mayella Ewell of raping her.
5. I think when Calpurnia takes the children to her church in chapter 12, this is a significant event because the children see another part of town, another race of people, they meet Rev. Sykes and observe how upsetting the trial is going to be for people other than their family. It is an important chapter for the racial theme of the novel. The children feel uncomfortable when they are the only whites, and they also notice that Calpurnia speaks and acts differently among her people than when she is working in their home. There is a lot that happens in this chapter, but the main thing is the trip to the other side of town represented by the visit to Cal's church.

I am sure others will have some different choices. What do you think? You can read the outline here on eNotes and see what you think.

You already have some great ideas from the other editors, but I want to help you consider the top two which I believe take place in chapters 10-11:

1. The situation with Mrs. Dubose demonstrates Atticus' reasons for taking the trial. He wanted the kids to learn two lessons from her. First, you don't just give up because you know you are going to loose the battle. She didn't give up, she tried to die free from her addiction. This will come into play in the trial. Second, be courageous enough to do the right thing. Sometimes, the reason people don't do the right thing is because it is hard. She did what she did because it was right.

2. The kids gain tremendous respect and learn of Atticus' incredible humility and humble nature when he shoots the dog. In this situation, he earns their impression of "cool", and he protects the entire neighborhood which I believe is symbolic of what he tries to do for Maycomb.

In To Kill a Mockingbird, the five (5) most important events before chapter 13 are, in order of importance:

1) Atticus agrees to defend Tom Robinson.  This causes anxiety for the family: Scout gets in trouble at school.  Aunt Alexandra will have to come to live with them.  This event sets up the entire second half of the book.

2) Jem nearly gets shot by Nathan Radley as he retrieves his pants.  Jem nearly dies, but it shows how desperate he is not to disappoint his father.

3) Scout begins school.  Her first grade year is spent spinning her wheels.  She learns nothing and only gets in trouble for trying to help.  The education system is backwards, much like the social outlook of the town.

4) Arthur Radley takes an interest in the kids, and vice versa.  They try to get him to come out, and he does: he leaves soap carvings for them.  Little do the know that he will serve as a protector for them.  He is the grey ghost who will save Jem and Scout at the end.  Foreshadowing.

5)  Jem and Scout go with Cal to her church.  The kids see how tight the black community is as they raise money for Tom Robinson's wife.  This field trip is invaluable: the kids learn more from Cal than they do in any other setting up until now.

Ill give you a few extras and let you decide for yourself.

• CHAPTER 1.  Dill arrives in Maycomb for the first time.
• CHAPTER 2.  Scout has a rough first day of school.
• CHAPTER 3.  Walter Cunningham has lunch with the Finches, and Atticus offers his "climb into his skin and walk around in it" advice.
• CHAPTER 4.  Jem and Scout get their first gifts from the knothole.
• CHAPTER 5.  Atticus catches the children trying to leave Boo a note.
• CHAPTER 6.  The children sneak onto the Radley porch and Jem loses his pants.
• CHAPTER 7.  The knothole is mysteriously sealed with cement.
• CHAPTER 8.  It snows, the kids build a snowman, and Miss Maudie's house burns down.
• CHAPTER 9.  Christmas at Finch Landing: Scout fights Francis and gets spanked by Uncle Jack.
• CHAPTER 10.  "One-Shot Finch" returns to kill the rabid dog.
• CHAPTER 11.  Jem serves his punishment with Mrs. Dubose, who dies at the end of the chapter.
• CHAPTER 12.  Jem and Scout attends Calpurnia's church.