What does the author persuade his reader to think, feel, or do?

First, one should note that the author of Pride and Prejudice, Jane Austen, was a woman. The work itself is a novel, not a polemical tract, and thus is not really intended to persuade anyone of anything. Its purpose was primarily to entertain its readers and earn money for its author.
That being said, the narrative voice of the novel does have a fairly strong point of view, as does the protagonist Elizabeth Bennett. In this period, arranged marriages were considered normal among the gentry. While there is no indication that the narrator believes in marrying on a whim or rebelling against social conventions, as we see in the disaster of Lydia's impulsive behavior, nonetheless the narrator and Elizabeth both advocate mutual respect and friendship as essential parts of a married relationship, and thus justify Elizabeth's refusal to marry Mr. Collins, although both are eventually reconciled to Charlotte Lucas' rather different decision. 
Austen's narrator values intelligence, self-discipline, and taking time to develop genuine knowledge rather than acting on the basis of superficial prejudices.