Dragon Ball/Dragon Ball Z Anime Story...

I'm pretty sure that the manga and anime versions are the same so both the manga and anime stories at VA are the same.

Dragon Ball is about a boy, named Goku, who has to search for all seven of the Dragon Balls. He has the fourth ball already since his grandfather left it for him. During his journey, his gains ownership of a golden cloud, that he can only ride. He gets help from Bulma and other fighters throughout the series. All of them have been in tournaments to earn money and to train.

Dragon Ball Z is about when Goku is older. He is still searching for the seven Dragon Balls, but for a different reason. Sometime in the series, Goku dies and becomes an angel. In the end, he does find all of the Dragon Balls and makes one wish.