What lessons has Sylvia learned?

By the end of the story, Sylvia says, "ain't nobody gonna beat me at nuthin." She seems to have internalized the fact that she is going to meet with opposition in society as a result of her race (and, perhaps, even her sex). There are people who can afford to spend a thousand dollars on a toy sailboat, and there are people who cannot. There are people who have a better chance at being able to afford such a thing someday, and there are people who do not. As a young black girl who seems not to be receiving an adequate education from her school, likely due to a lack of sufficient funding for public schools with a primarily black population, Sylvia will likely meet with a great deal of opposition in her life.
When Sugar, Sylvia's friend, comes to a realization that America is "not much of a democracy" and that individuals of different races do not have an "equal crack at the dough," Sylvia can feel something "weird" going on "in [her] chest." Miss Moore looks directly at Sylvia and asks if anyone else has learned anything today. I'm not sure that Sylvia is really aware of what she has learned, perhaps, when she is a child, but her defensiveness seems to indicate that her eyes have been opened to a new understanding of the world.

After an educated black woman, Miss Moore, takes a group of black children from a poor neighborhood to see FAO Schwartz, a high-end New York toy store, Sylvia has much to think about. She understands the lesson she is supposed to take away, which is that black people are treated unjustly in America and should demand their share of the "pie," but she resists falling into line behind someone else's agenda. In other words, she holds onto her desire to think and analyze for herself. In fact, she states,

I got a headache for thinkin so hard.

She repeats a second time that

I’m going . . . to think this day through.

At the end of the story, Sylvia has not processed all she has seen, but she shows an important openness to learning and changing. One of the issues Sylvia has to contend with is that, though she is also black, Miss Moore (who has "more") is not in the same place as the black children she is trying to reach. Her education, political radicalization, and higher place on the class ladder have separated her from the very people she wants to reach.
Sylvia is well aware that white people have more than black people, a lesson reinforced by her field trip to the toy store. But she is not sure what the "pie," which she is supposed to demand more of, is, exactly. Her thinking, as a child's would be, is concrete, focused more on the reality of the here and now than on abstract political or economic theory. She knows, for example, that Miss Moore is not that smart, because she never retrieved the change from Sylvia's taxi ride. Sylvia plans to spend it with a friend on cake and other treats. That, to her, is a win with a concrete reality, which social justice does not have.
Sylvia needs to think more about what her place in the world should be. But she is thinking, and she decides,

ain’t nobody gonna beat me at nuthin.

That, at least, is the beginning of an assertive lesson that Sylvia will presumably refine over time as she gains a more sophisticated knowledge of what not being beaten means.

At the end of “The Lesson,” Sylvia seems to be having the most difficulty of all the children in processing the day’s experience. The author presents her as probably the most intelligent and potentially the most insightful in the group, but she resists the interpretation that she intuits that Miss Moore is encouraging in them. Although the young woman has good intentions in exposing the poor children to the vast gulf between their purchasing power and the expensive toys in the store, Sylvia seems to have sensed her aura of condescension.
Sylvia’s attitude toward Miss Moore is as important as her attitude toward the racial and social inequalities evident at the store and on their field trip more generally. While she accepts Miss Moore’s message of the importance of desiring social change, she is skeptical about her own possible activist role. In some regards, this young woman is just another authority figure, and her elite education distances her from the children whose consciousness she aims to raise. When Miss Moore declaims about change, Sylvia notes, she then “waits for somebody to say that poor people have to wake up and demand their share of the pie.” Sylvia’s inchoate sense is that by putting words into their mouths, Miss Moore is encouraging them to say things they do not understand: "none of us know what kind of pie she talking about in the first damn place.'' It seems that Sylvia will continue trying to learn about the pie and decide for herself how much of it she desires.