Vegeta is as arrogant as the mountains under the ocean are high and an ego as long as the distance between the moon and sun. He thinks he's the best of the best because he is prince. He doesn't understand how Goku can be more powerful.
Vegeta comes to Earth to take it over and sell it, which gets all the Z warriors involved. After the epic showdown between Goku and Vegeta, Goku lets him go despite Krilin's warning to kill him. Goku proves that he is right when Vegeta helps them on Namek later.
His quest to beat Goku becomes one of the long running threads in the series. It goes from pride's sake to one where he wants to push himself farther.
He returns to Earth, still cold and snotty and ruthless. He slowly mellows as time goes by. He even becomes a family man. A hardened warrior going to an amusement park with his son. Yeah, not something I pictured either but it happens.
