Not only does the play celebrate the wonder of the discovery of a "brave new world," it shows the characters in the play making discoveries about themselves and others.
For example, at the end of the play Caliban discovers that Stephano is not a god. In a moment of insight that perhaps reveals he has more depths than Prospero perceives, Caliban refers to himself as a “thrice-double ass” for his mistake.
Alonso discovers a sense of remorse over what he has done to Prospero, even if Antonio does not.
Prospero has been deeply shocked and hurt by the treachery of Antonio. He had thought he could he could trust his brother, if nobody else, and the fact that his own close kin turned on him made his sense of injury all the greater. Nevertheless, by the end of the play, Prospero discovers he can forgive his enemies, even if neither Antonio nor Sebastian feel remorse. Prospero states:
The rarer action is
In virtue than in vengeance.
In other words, it would be commonplace to take revenge on his enemies, having them in his power, but he is choosing the more unusual path of "virtue" by showing mercy and forgiveness. This ends the play on a positive note, and sounds a theme important to Shakespeare, which is that mercy is an all-important trait, especially in powerful people. What we discover inside, Shakespeare implies, is more important than all the new worlds we might find.