In order to understand why Scout is picking on Walter Cunningham at the beginning of chapter 3 of To Kill A Mockingbird, the reader must first look at the events that occurred previously in chapter 2. In chapter 2, the reader is introduced to Walter Cunningham in a scene where Scout's teacher, Miss Caroline, asks everyone not going home for lunch to show her a lunch pail. It is in this scene Miss Caroline stops to ask Walter where his lunch is. After Walter tells Miss Caroline he forgot his lunch, she offers him a quarter to go and buy lunch for himself. Because Walter comes from a family with little money, and he does not want to be in debt to Miss Caroline, he refuses. Miss Caroline insists, and her and Walter go back and forth about his taking of the money until Scout interrupts. Thinking she is doing the right thing, Scout tells Miss Caroline about how Walter's family is from the country and poor, and therefore he can not afford to take the money from Miss Caroline as he would not be able to pay her back. It is during this time we learn that Walter's father was once in debt to Atticus and had to pay him back with stove wood and other goods for his services. After speaking up on Walter's behalf and telling Miss Caroline "You're shamin' him" in front of the entire class Scout is summoned to the front. Miss Caroline tells her "You're starting off on the wrong foot in every way, my dear." She is then disciplined by Miss Caroline in the form of by being struck on the hand with a ruler and told to stand in the corner. This leads to the class laughing at Scout as she is punished for doing what she believed was the right thing.
Because all of this happened and Walter did not say a word, Scout begins chapter 3 by picking on Walter in the schoolyard. It is only through Jem's intervention that Scout quits. It is here Scout tells Jem why she was picking on Walter when she says, "He made me start off on the wrong foot." This is a reference back to earlier in chapter two when after Scout is talking about Walter to Miss Caroline, Miss Caroline comments "You're starting off on the wrong foot in every way, my dear" before punishing Scout. Because she uses similar wording to Miss Caroline, the reader can infer that this is indeed why Scout though it necessary to pick on Walter once class let out: because she felt he was responsible for her rough start in school. Hope this helps!