Catherine travels from Fullerton to Bath to Northanger. Elinor and Marianne travel from Norland to Barton to London to Cleveland. Why do the heroines have to leave their homes? How do their geographic locations connect to plot and character development?

All three heroines have to journey from the sheltered safety of their birth homes to learn, suffer, and grow. Catherine is sheltered and naive when she arrives in Bath under the protection of an older family friend. At first, Catherine takes the people she meets at their word, for example, not comprehending the manipulations and hypocrisies of her new "best friend" Isabella Thorpe. Much of the comedy of the first part of the novel hinges on Catherine's unworldliness and the dramatic irony it produces. Dramatic irony is when the the audience knows what a character in a work of literature doesn't, and in the case of Northanger Abbey, we as readers quickly understand that Catherine is being used by people who don't really care about her. Catherine gradually wakes up and learns to discern the difference between words and deeds. This enables her to mature and become a worthy companion for Henry Tilney.
Marianne in Sense and Sensibility is likewise an innocent, sheltered by a comfortable life, until the death of her father throws her, her mother, and her sisters on the world. Like Catherine, Marianne is initially deceived by words, in her case the language of romanticism, or sensibility. She believes that money doesn't matter, that people should be judged based on how much emotion and sensitivity they express, and that true love will overcome all obstacles. When she travels to London, which like Bath, represents the colder, harder values of the "world," she realizes that her romantic dreams are merely dreams: Willoughby, her "true love," marries another woman for money. It is only by journeying outward and learning to face reality (which almost kills her) that Marianne can enter into marriage with the less dashing but more stable Colonel Brandon.
Finally, Elinor, "sense" in Sense and Sensibility is treated scornfully as a poor relation by her Dashwood in-laws when she travels to London, and must also endure the deceit of Lucy Steele. She learns that instead of suffering in silence through all her troubles, she might have done better to open up more to the people around her. Her sufferings, as with the other characters, increase with her travels, and from these sufferings she learns and matures.