The story begins on a planet named Vegita where there is a race of people called the saiyans. They are a very powerful race of people that turn into monkeys when the moon comes out. The planet gets destroyed by frieza, a bad alien. Everyone on it dies except for a handful of saiyans and two of the main characters. One is Goku and the other one is prince Vegita. Goku, our main hero, ends up on Earth and is found by Gohan, a human man. He takes care of Goku. They live in the forest and stuff. Then one day Gohan dies and Goku finds a human girl named Bulma. She's around the age of 16 or 17. Goku, being just a kid (around 10 years old) and living in the forest all his life, dosen't know who or what she is. Then they start to talk and become friends. Bulma tells Goku that she is on a quest to find the dragon balls. If all seven are collected then a dragon comes out and grants one wish. Goku also trains martial arts with them. Then they both go on the quest, and they eventually find Master Roshi and his student Krillin. On one of their quests they meet Oolong, a shape shifting pig. On another on they meet chi chi. She is the daughter of the great Ox-King. Then they find more more friends like Yamcha and Bwar which are two desert pirates. They also meet up with Chiaotzu and Tien who are a pair of warrior monks. They also meet Piccolo who is a Namekian. He is green and has antennae. He is evil at first, but then he splits, creating the new gaurdian of Earth and Piccolo the lovable green guy. Then they all go on the quest to find the dragon balls and stuff, and they try to prevent evil people from getting them like the red ribbon army and Emperor Pilaf.
The story of Dragon Ball Z starts off when the team from Dragon Ball (meaning Bulma, Yamcha, Puar, Tien, Chioatzu, Piccolo, Krillin, Chi-Chi, and of course Goku) are all much older. Goku married Chi-Chi and had a son named Gohan(named after his grandfather). The story is broken up into sagas in which they fight evil and try to keep the universe safe. One of the first ones is the saiyan saga were they meet Vegita (who is evil) and they defeat him. Then he becomes their friend in the Frieza saga were they defeat Frieza who is working his way around the universe destroying stuff. Then there is the android saga were an evil doctor/scientist created some androids tto destroy the world. They kill them all except for #18, who is in love with Krillin. Then they fight Cell who was also created by the evil doctor/scientist and was made to eat the androids to become more powerful. Goku dies there and we find out that he leaves behind another son besides Gohan, and his name is Goten. Vegita merried Bulma and they had a son named Trunks. Then some 15 years later there is the Buu saga. They have to defeat Majin Buu but first he practically kills everyone and everything. Then they bring Goku and Vegita back and Buu transforms about 8 different times. Each transformation makes him stronger. Goku eventually kills him, but Buu comes back in a good version whose name is Uub. Then Goku goes off to trian him. Gohan marries Videl, who is hercul's (AKA Mr. Satan) daughter, and they have a daugher named Pan. Hercul (Mr. Satan) is the best wrestler on earth.
For the first 20 or so episodes Goku and Pan (Gohan and Videls daughter) fly around trying to find the black star dragon balls becouse emporer pilaf used them on Goku, making him small. Also if they don't find then the earth will be destroyed. Thats all I've seen so far.