In addition to the response above, I think it contributes to painting a picture of the class system in Maycomb and Scout's rejection of the status quo, both of which will be integral parts of her understanding of the trial.
Scout's school experience introduces us to the Cunninghams, the Ewells, the bus children and how they compare with Scout intellectually. Scout, far superior, still has moral values and sees how the aforementioned gimmicks are truly a rip-off.
Scout is not okay with the system of school, nor will she later be okay with the system of racism when she watches it unfold in the trial.
Additionally, I think Lee takes this school experience chapter to stab at the problems of public education... ironically many of those same problems still exist today. We try to serve the needs of a diversely growing group of children in a society where information grows faster than our teachers.
I think it is also especially relevant that Lee is taking on the educational system as well as the class and race situation in her novel. It is interesting to note that even today education is based on theory that is often not as good in practice as it is on paper. Scout bucks the system in many ways. She is a nonconformist in a society that asks us to conform or else. Her first day at school emphasizes this because she is far more advanced and far more worldly than her classmates and even than her teacher. She understands poverty, to an extent, better than Miss Caroline does (although she does not really gain a true understanding of what she knows until the story progresses). Ultimately, Scout learns that you have to learn how to live within the system even as you seek to change the system.
Harper Lee seems to have two purposes in presenting such a detailed description of Scout's first day of school. First, she uses the event to provide more information to the reader about Scout's character. We discover that Scout is independent (especially when she is away from Jem) and that she is precocious and extremely intelligent. We also find out from the scene that Scout struggles with how to respond to adults who are not open-minded like Atticus and Miss Maudie and who treat her like a simple child.
Secondly, Lee incorporates a satire of the education system of her childhood in order to shed light onits flaws. Miss Caroline represents not only a new teacher's tendency to try different "gimmicks" with her students but also her insecurity as a newcomer to Maycomb's culture and to education in general. Lee's underlying attitude toward the public school system of her day is ridiculing and ironic. She seems to present education as holding back inquisitive, high-achieving students (hence, Miss Caroline's telling Scout not to read) and as being homogeneous in its methods--teach all students the same no matter their home environment, learning styles or abilities.