Carrie's full name is Caroline Meeber. Sister Carrie is a nickname her family gave her.
Carrie lets other people, especially men, make decisions for her. She has an affair with traveling salesman Charles Drouet when she is very young. She leaves Drouet for his friend George Hurstwood. When Hurstwood's wife finds out about the affair, she starts divorce proceedings and threatens to take every penny from him. Carrie and Hurstwood run off together and pose as a married couple under an assumed name. So Carrie becomes Mrs. Wheeler.
Life as Mrs. Wheeler is not what she thought it would be, however, and Carrie decides to look for work in the theater. She joins a chorus line under the name Carrie Madenda.
So you have Caroline Meeber, Sister Carrie, Carrie Wheeler, and Carrie Madenda.
By the way, there's a great movie version of the story starring Jennifer Jones and Laurence Olivier.
The names for Carrie are exactly correct. However, I don't think that Carrie "let other people, especially men" make decisions for her. It's even questionable about whether anyone "makes" any decisions for themselves in the novel. Carrie and Drouet are both enamoured of much the same things: clothes, money, the trappings of "the good life." Carrie is OK with Drouet until someone better comes along in the person of Hurstwood, and then she "switches" her allegiances for the additional glitter that he can bring to her life. Hurstwood's somewhat miserable life (his wife is pretty awful, not to mention his children) drives him in the search for something else. He knew divorce would be the end of his present life, but he seemed unable to deny an almost animal attraction ... almost?
All these characters are driven by what Dreiser called "chemisms." Rather than making "decisions," they are like the moth attracted to the flame, an almost senseless attraction, but one that is inexorable.
Also, Carrie is shortly mrs. Murdock , when Hurstwood chooses that last name for them, but he changes it to Weeber because Carrie doesn't like it. This happens on they way to New York city.