Except for me, no one seemed a stranger to anything or anyone. Hearing the way their greeted each other, I couldn't be sure that they hadn't all come out of the same woman's belly, and at the same time, too. (2.2)
Annie is in her characteristic observation mode, taking in the sea of new faces on her first day of school. She feels different from all the others. Interestingly, she relates this to the figure of the mother when she says she can't be sure they "hadn't all come out of the same woman's belly." Annie assumes here that you can learn a lot about a person by knowing their mother.
I didn't exactly tell a lie about the last part. That is just what would have happened in the old days. […] But the real truth was that I couldn't bear to have anyone see how deep in disfavor I was with my mother. (3.14)
When Annie speaks about her invented conclusion to her "autobiographical essay" she wrote on the first day of class, she says she "didn't exactly tell a lie." Annie is, from a young age, exploring the genre of fiction.
I was soon given responsibility for overseeing the class in the teacher's absence. […] I indulged many things, especially if they would end in a laugh or something touching. I would never dillydally with a decision, always making up my mind right away about the thing in front of me. (3.19)
As Annie is finding her way in the world and making new friends at the school, she explores, experiments and "indulges" in a lot. She's a very intelligent class clown, always good at cooking up something either funny or moving.
At recess, among the tombstones, I of course had to exhibit and demonstrate. None of the others were menstruating yet. I showed everything without the least bit of flourish, since my heart wasn't in it. I wished instead that one of the other girls were in my place and that I were just sitting there in amazement. (3.26)
After Annie starts menstruating, she shares what it's like with her buddies at the tombstones. The tombstones represent the main space for exploration and sharing secrets.
Then, after a proper time had passed, I would quietly unlatch the gate, creep back into the yard, and dive under the house to extract or hide some object that was forbidden me—usually some object that had come into my possession through my expert stealing. (4.1)
Stealing is a not-so-great aspect of Annie's exploration. She tests the boundaries of her relationship with her mother and the rules her mother enforces by stealing what are essentially useless pieces of junk.
Ruth I liked, because she was such a dunce and came from England and had yellow hair. When I first met her, I used to walk her home and sing bad songs to her just to see her turn pink, as if I had spilled hot water all over her. (5.2)
Annie loves the novelty of new things. That's why she takes an interest in Ruth who is from England with yellow hair: Ruth is a new character for Annie, and Annie is fascinated with both Ruth's life story and how best to mortify her.
Of course, sometimes, what with our teachers and our books, it was hard for us to tell on which side we really now belonged—with the masters or the slaves—for it was all history; it was all in the past, and everybody behaved differently now. (5.5)
Annie is clearly a thoughtful and inquisitive girl. She never takes her history lessons at face value and wonders where she fits in the world in the big scheme of things.
I was no longer on the same chapter we were studying. I was way ahead, at the end of the chapter about Columbus's third voyage. (5.6)
As a great student, Annie soars ahead of her friends and loves to explore her studies. She moves far beyond the other students in her class because she always wants to know more.
When I next saw the picture of Columbus sitting there all locked up in his chains, I wrote under it the words "The Great Man Can No longer Just Get Up and Go." I had written this out with my fountain pen, and in Old English lettering—a script I had recently mastered. (5.6)
Sometimes Annie's exploration leads her to down the wrong path. The defacement of the Columbus picture, while funny (and insightful), gets her in a whole heap of trouble.
Then all the sound rocked back and forth in my ears, and I had a picture of it; it looked like a large wave constantly dashing up against a wall in the sea, and the whole thing made me feel far away and weightless. I could hear the rain on the roof, and it was still pinning me down. I looked inside my head. A black thing was lying down there, and it shut out all my memory of the things that had happened to me. (7.6)
In addition to exploring the things around her, Annie also explores her inside thoughts and dreams… including her darker side (helpfully personified as a "black thing.")