Why does Betty Parris start accusing people in the first act?
The audience knows that Betty is essentially faking her illness (or at least its severity) because, when all of the adults have left the room, she awakens in order to confront Abigail about the evil thing Abigail did in the woods the night before: drinking a charm to kill Elizabeth Proctor, the wife of Abigail's former lover. Abigail did not confess this to her uncle, Betty's father, and Betty knows this.
No adult present has seen Betty behave at all coherently until she rises from her bed, having heard Abigail's accusation of Tituba as well as Tituba's own confession to witchcraft. Hearing such a confession may be quite compelling for young Betty; how is she to know that Tituba confesses only to protect herself? Once Abigail adds her own accusations to the one Tituba makes, Betty rises with "a fever in her eyes, and picks up the chant." It is possible that Betty, maybe, has realized the same thing Abigail has: by accusing others, they deflect attention from themselves. It is also possible that all of the tension Betty feels as a result of her fear is released when she begins to accuse others. After all, she is a child, and she is likely terrified by the events she witnessed in the woods and by Tituba's own confession. If we can believe that there is any possibility that Betty does not intend to hurt others, then it seems possible that she is simply looking for a way to feel better herself, and making accusations is one way to do that because Reverend Hale has promised Tituba that the town will glorify her for helping to cleanse it of evil. There are many possible reasons for Betty's accusations of others at the end of this act, and it really depends on how conscious we believe an eleven-year-old can be of the consequences of others' and her own actions.
Betty starts accusing people to deflect suspicion away from herself and the other girls from their dabblings in the occult in the woods.
She is terrified into submission by Abigail, who tells Betty and the other girls, "...Let either of you breathe a word, or the edge of a word, about the other things, and I will come to you in the black of some terrible night and I will bring you a pointy reckoning that will shudder you. And you know I can do it...".