Lets go back a little...
Just like everything else, Goku started out as a baby. His name was Kakarot. He was born on Planet Vegeta, just a day before it was destroyed.
He was sent to earth right before Frieza destroyed Vegeta. His father, Bardock, fought boldly until the end.
Bardock
Frieza
If it wern't for a passing stranger (and a terrible head injury), the young saiyan would have (believe it or not) destroyed the earth and grow up to be insanely evil!! Grampa Gohan changed his name to Goku.
Goku's childhood is pretty much all of Dragon Ball. And I'm not going to tell you EVERYTHING that happened in that show so I'll summarize it...
His adoptive grampa continues to raise him until he died, so Goku had to raise himself. But he also had Gohan's prized possesion, that he would later learn was the four star dragonball.
A rematch came when an adult Goku had to fight King Piccolo's son, Piccolo Jr. or just plainly Piccolo. Goku won, but Piccolo still survived
That kinda wraps it up for Dragon Ball, now let's move on. Dragon Ball Z is mainly about Goku's fatherhood. In some parts, people think that the main character switches to Gohan, but remember: it's ALWAYS Goku.
5 years has passed since Goku has fought Piccolo and settled down. And Goku and Gohan are going to visit some old friends of Goku.
...And a new enemy, a surviving saiyan named Raditz, who tells Goku of his saiyan past and how he's Goku's older brother. Goku doesn't believe a word of it.
Then he tells Goku to join him and two other surviving saiyans to kill natives of habitable planets and sell the planet to people who need them. Goku never wanted a job like that, but Raditz won't take no for an answer.
So he kidnaps Gohan and holds him for ransom. Goku's options are limited. He wasn't strong enough alone to fight his "brother".
So he joins forces with Piccolo to stop Raditz and save his only son. They were still unable to beat him. After Gohan sees Raditz hurt "Daddy", his power raises and, just for a moment, becomes stronger than Raditz, and uses his anger-controlled-strength to attack Raditz and injure him. Then his power level drops back to zero when he gets scared of Raditz.
This is when you ask, "Is that it?! Did I click on that prezi for THAT?!" Fortunately, no believe or not. There is still more.
Before Goku and Raditz died, Raditz tells Goku and Piccolo of two more saiyans that will come in a year's time. The Z Fighters begin to plan...
Piccolo, predicting that he might die in a year, decides to train Gohan and control his raw saiyan power. Meanwhile in other world (religous people, don't get offended, it's just a show), Goku trains with King Kai after traveling on Snake Way. And Krillin, Tien, Chastou, Yamcha, and Yajarobi train with Kami and Mr. Popo, all preparing for the Battle of The Saiyans.
A year has passed, Goku is wished back to life with the Dragon Balls. To shorten up the Battle of the Saiyans, Yamcha, Tien, Chastou, Piccolo and Nappa (saiyan) dies. Vegeta (other saiyan) barely lives and escapes.
Hope still remained on Earth after the fight, but because of the terrible loss, they had to use the Dragon Balls on Planet Namek (Kami's home world). Goku couldn't come because he was too injured from the battle. But when he was well enough, he found that Vegeta was at Namek too. He also fought The Ginyu Force, en elite, well-choreographic team of fighters who worked for Frieza. He struggled most when fighting the team's captain, Ginyu, who switched bodies with him using a special trick.
Goku
vs.
Frieza
Not too long into the battle, Goku realized he might lose and die. He was on the verge of surrender when Vegeta's spirit told him not to give up, to remember what Frieza's done, and what he will do if he doesn't beat him. Goku has one last trick up his sleeve...
But somehow Frieza survived it, and he killed Krillin and almost killed Piccolo (he was wished back to life with the Namekian Dragon Balls). Goku snaps. His anger transforms him into a saiyan legend that Frieza has feared for years... a super saiyan!
Now he's stronger than ever, but he also wasn't the same. He was more serious and filled with anger. But he knew his purpose, to kill Frieza!
In the end, he won. If his father were there, he would be so proud of Goku.
Ssj3
10 years after Goku kills Kid Buu, his son, Gohan is married and has a child of his own. A 4 year old daughter named Pan. Goku's a GRANDFATHER!!!
In a World Martial Arts Tournament, Goku meets a 10 year old Uub, the reincarnation of Buu. Uub becomes Goku's pupil and the two leave unplanned for 10 years to train.
Dragon Ball Z ends and Dragon Ball GT begins ten years later. GT stands for "Grand Tour" if you were wondering. GT is the only show of the DB trilogy that doesn't start from a manga, so it's non-canon.
Goku is accidentally wished back to a kid. Now he, Trunks (now 28 years old), Pan (now 14), and Giru (a new addition to the team) are on a quest to find the black-star dragon balls that had spread out throughout the galaxy!
Along the way, the saiyan trio and their robot companion come across an evil scientist called Dr.Myuu. He reveals his creation, Baby, which ends up mind controlling almost everyone on earth. Baby picks Vegeta for his new form and becomes Baby Vegeta.
Dr.Myuu
Baby
Baby Vegeta
Goku and Pan's family reunion came to a shattering end when they realize Chichi, Videl (Pan's mom), Gohan, and Goten's strange behaviors.
Pan's own father and uncle (Gohan and Goten) fought her until she was unconscious.
Because of that, Goku thought his children had became demented! He was about to make the difficult decision of KILLING his own sons!
UNTIL...
Goku's brother-in-law (Videl's dad), Hercule Satan, came with his friend, Majain Buu. He told Goku that his sons and the rest of the world were being controlled by Baby.
The series ends with an adult Goku walking out of the tournament and reminiscing on his past adventures. Afterward, Goku looks at the audience, smiles and flies into the air with his power pole, and says, "Until we meet again, you guys!"
Not too long after, Goku marries his "childhood sweetheart", Chichi. Together they lived in peace and even had a son, Gohan (named after Goku's adoptive grandfather).
Goku Son
After about a year of training, Goku had to defeat the evil King Piccolo who had a wicked vision of taking over and controlling the world. Let's just say he had to...step down :)
He also met another boy named Krillin who became his best friend, and an old wise martial artist named Roshi. He became Goku and Krillin's master.
When he was 12, a teenage girl named Bulma changed Goku's life forever!
Goku ends up saving the world and kills Baby. He continues fighting against bad guys. Goku and Vegeta both found a fourth level of super saiyan to help them win their battles.
Gohan
Goten
When Goku, Trunks, Pan, and Giru returned to earth, they gave the Black-Star Dragon Balls to Dende (the new guardian of Earth) although they did not know he and Mr.Popo were already controlled by Baby. They return home leaving Goku as a kid to grow up on his own.
Warning!
This next part may not be suitable for some viewers
(just skip one slide if you don't want to see it)
After Goku kills Omega Shenron, he sees that the dragon balls become dangerous when being used too many times. He decided to leave with Shenron and continue his training elsewhere. It seemed he had dissappeared from the world for a hundred years...
(Back left to right) Recoome and Burter
(Middle left to right) Guldo and Jeice
(Front) Captain Ginyu
After recovering from that battle, Goku finally meets Frieza, although he was too late to save Vegeta. Before he died, he told Goku (he calls him Kakarot) how it was Frieza who destroyed their home planet, Vegeta. Goku buries Vegeta and faces his opponent in what will be the most historical fight in DBZ.
Out of his own decision, Goku sacrifices his own life to put an end to the saiyan. Piccolo fires his Special Beam Cannon...
Goku also now has a second son named Goten. He is best friends with Bulma and Vegeta's son, Trunks.
As Goku gets stronger, he fights more and more enemies. Some including Broly, Cell, and Buu. He manages to reach a super saiyan level 3! (ssj3)
A hundred years later, Pan's grandson, Goku Jr. is a super saiyan, and he fights against Vegeta Jr. in a World Martial Arts Tournament.