Goku is the principal hero of the entire "Dragon Ball" saga, appearing in the original "Dragon Ball" as a child, "Dragon Ball Z" as an adult, and "Dragon Ball GT" as an adult transformed into a child. Vegeta first appears in early installments of "Dragon Ball Z" as a villain but eventually changes sides.
Originally from the planet Saiya, Goku was sent to Earth on a mission to destroy the planet but suffers a head injury and loses all memory. As he grows up into a powerful but noble warrior, he marries the Ox King's daughter, Chi-Chi, and has two children, Gohan and Goten.
Vegeta is not only the prince of the Saiyans but also among the last of the race. He initially appears on Earth to sell the planet after wiping out all life on it. Later, when the characters travel to the planet Namek, he seeks that planet's Dragon Balls to gain immortality.
While Vegeta is small in stature, he is very powerful and ambitious, which leads to his goal of surpassing Goku in power. Conversely, Goku is extremely powerful but also extremely trusting and naive. Both are capable of transforming into golden-haired "Super Saiyans," of which there are multiple levels, with each subsequent level being more powerful than the last.
Soon after his initial appearance, Vegeta decides to fight on Goku's side during their adventure on Namek. Soon after, he forms a relationship with the character Bulma and fathers a son, Trunks, who also grows up into a powerful warrior, and a daughter named Bra.