In chapter 14 of volume 1, called "Secrets," we learn of Jo's dreams:
to be independent and earn the praise of those she loved were the dearest wishes of her heart.
In this chapter, she is delighted to have her dream of being a published writer come to fruition as her short story "The Rival Painters" is printed in a local paper. She tells her sisters and Marmee,
I shall write more ... and I am so happy, for in time I may be able to support myself and help the girls.
Jo does write and publish more and is able to earn money this way, achieving the independence she dreams of.
However, as the first quote shows, Jo wants more than simply to be a writer. Being a writer is something she enjoys, but it is, more than anything, a means to the greater goal of financial independence. Second, her two dreams—independence and "the praise of those she loved"—are often at odds. Just as she is gaining success as a writer, Jo is also getting the first ominous sign, indicated to her by Laurie, that John Brooke is in love with Meg. Jo's dream is to have her family around her, but that dream will be threatened and undermined as her sisters marry, die, or disappear abroad, leaving her lonely. Her writing, too, is undermined when Mr. Baer criticizes her for wasting her talent writing potboilers that aren't worthy of her.
Jo, however, achieves a version of her dream by the end of the novel by making compromises. She marries Mr. Baer and with him embarks on opening a school in a big house she has inherited from her Aunt March. This way, she will contribute to earning a living and draw a new family around her.