Gohan is a young and cheerful kid. In the beginning of DBZ, he was naive in thinking that he will be helped with everything. Gohan is more powerful than Goku in every aspect except emotion. Unlike his father, Gohan wants to stay and help instead of escape when an enemy is much stronger than him. Gohan has a lot of hidden power and can involuntarily call upon it when he is faced with a situation beyond his control (He explains that in the Cell Game). When Piccolo trained him in the wilderness after the Raditz Saga, he carried around a sword that he used to defeat certain enemies. Gohan is plenty strong...
Throughout the events of DB, Goku never knew he was a saiya-jin. He was the most powerful "human" with a tail in the entire DB series. When he was youg, he fell into a ravine, hit his head, and forgot his original mission to destroy inhabitants of Earth and sell the planet for money. Goku turns to the side of good and is raised by his foster grandfather, Son Gohan (in Asian culture, the last name comes first) and so Kakarot is then dubbed Son Goku. He meets the rest of the DB cast later and alone, chalenges the entire Red Ribbon Army and wins, leaving Dr. Gero to live (Get the RR on the androids?). A strong warrior indeed...
Vegeta is part of royal saiya-jin blood, which has a chance of its bloodline to be super saiya-jin. Vegeta, when he was young, was given to Freeza to escape Planet Vegeta's (what a coincedence!) horrible fate. When he was young, he was cruel and meeting Freeza made it worse. Vegeta develops a hatred for Freeza and everything around him and desires only one thing, ultimate power. But since Freeza was already more powerful than Vegeta, he wanted immortality to fight Freeza and win. To Vegeta's surprise, a lowly saiya-jin by the name of Goku turns into a super saiya-jin and defeats Freeza. This awakens his thirst for competition...
