I can think of two ways in which discovery gives characters greater knowledge of society and themselves in Shakespeare's The Tempest. Miranda's discoveries of the existence of other humans leads her to develop both a knowledge of her place in romantic relationships and a more earnest love for human society, while Caliban's discovery of human language leads him to more fully comprehend the bitter realities of hierarchy.
To begin with, it's important to understand that Miranda grew up on the island and only has relationships with her father, Prospero, and their wretched servant, Caliban. As such, she has no knowledge of what it's like to be in a romantic relationship. Meeting Ferdinand (the only "marriageable" male she's ever met) leads her to quickly develop an understanding of love and commitment. Further on, toward the end of the play in Act 5, Scene 1, Miranda meets even more humans, leading her to speak the now famous lines: "O brave new world that has such people in't!" (183-4). Here, we can clearly see that Miranda has developed a new excitement for the mere process of being human, now that she has discovered what it's like to live in a human society full of diverse individuals.
In contrast, Caliban's discovery is less positive. Caliban was the powerful son of the witch Sycorax before Prospero overthrew her and took over the island. When the play begins, we see him as a conquered native with little freedom. While Prospero taught Caliban language, this discovery only enables him to more fully understand his serf-like position in human society. He says, "You taught me language, and my profit on't/ Is, I know how to curse" (363-4). Through this statement, we can see that, through his discovery of human language, Caliban "discovers" his weak position within the hierarchy of human society, and curses it.