This is likely The Land of Elyon series by Patrick Carman. There are four books and a prequel. As you remember, she is initially approached by a rabbit in the library in the first book, and in the third book, she learns who her real parents are.
Summary excerpt of the first book, The Dark Hills Divide:
12-year-old Alexa Daley is spending another summer in Bridewell with her father, the Mayor of Lathbury. The book starts by Alexa going on a walk through the streets of Bridewell with her adventurer friend Thomas Warvold. During the stroll, Warvold tells Alexa a fable that he heard on one of his far-off journeys. When the fable is finished, Alexa finds that Warvold is dead.
Later on, Alexa uses the silver key to open a passageway hidden behind her favorite nook. At the end of the passageway, outside the wall, Alexa is greeted by a short man named Yipes, who seems to have been waiting for her. Alexa follows Yipes up Mount Norwood and comes to a glowing pond. Inside the pond, she finds a green stone. Once she puts it in her leather pouch, Yipes takes her to his house. Waking up from her sleep, Alexa finds that she has the ability to talk to animals. Darius, a wolf separated from his family, leads her to a tunnel under the walled road between Lathbury and Bridewell. At the end of the tunnel, Alexa discovers that the convicts of Ainsworth inhabit The Dark Hills. Once she is back outside, a rabbit named Malcolm takes her to the forest king Ander, a grizzly bear. Ander tells Alexa that she can talk to animals as long as she has the green stone with her and is outside the walls. He also tells her that the convicts living in The Dark Hills are planning to attack and take over Bridewell. The leader of the convicts is someone the convicts call Sebastian, an escaped convict posing as a citizen of Bridewell. At the meeting, Alexa meets Murphy (Squirrel), Beaker (Raccoon), Henry (Badger), Picardy (Female Black Bear), Boone (Bobcat), Odessa (Darius's wife) and Sherwin (Darius's son).
Summary of the third book, The Tenth City:
Alexa goes back to Bridewell to rescue her friend Yipes with Warvold, Armon, and the rest of the animals. After the death of Thomas Warvold, the brother of Roland Warvold, Alexa learns that Catherine and Thomas were her parents and her foster parents are Catherine's sister Laura and her husband James Daley.
For the sake of further reference, my search terms were book series young girl can talk to animals.