In Context: Goten was born in late March 767 instead of May. Since he was born prematurely. However, Gohan would be born in February 768 instead of Late December 767. Since he was born full term.
10 months after the 23rd tournament, Chichi has given birth to her first child. Although, the baby was born at 34 weeks. So, naturally, she became a bit nicer towards him. However, she still wants her son to become a scholar.
Goku was about to name him "Gohan". Until he saw how similar their hair is. He thought about naming him "Goku Jr" but then decided on "Goten". Since it's similar to Gohan; Ten means heaven as Ku means sky, so they connect; and Grandpa Gohan is in the afterlife.
When Goten was about 3 1/2, he started to feel curious about his dad's training. So Goku agrees to let him watch. Goten was just fascinated by it and wanted to learn some techniques. This made Goku smile and offered to do lessons for him.
As the two came back home, Goku told Chichi about their session. However, Chichi was not happy about it. In fact, she was planning to homeschool Goten once he turns four. This made the two confused. Chichi explained to them that Goten needs a high amount of knowledge and training would make him into a diligent. Goku was shocked by this because Chichi was a martial artist herself and even joined the 23rd tournament.
On Goten's fourth birthday, Goku has a four-star Dragon Ball necklace for him. Goten absolutely loves it and hugs his father. A day after that, Chichi started schooling Goten. However, it much more challenging for her than she excepted. Whereas, Goten has a hard time sitting still and doesn't like to study. This results in a lot of yelling and tantrums. Therefore, he and Goku would sneak off to train whenever Chichi goes out
When Goten was 4 1/2, Goku got an invitation from Roshi. So he decides to take Goten with him. Once, the two came over to Kame's house. Krillin, Bulma, and Roshi already knew that Goten is Goku's son. Since Goten is more sociable, he immediately told them about his name, age, and hobby. Krillin asked if he trains with his father every day. Goku gave out a sigh and said that Chichi has been making him study. So their training sessions are limited. Roshi laughed and said that Chichi has turned from a tomboy to a school mama. Bulma noticed that Goten doesn't have a tail. Goku simply said that he was born without one. Another thing she recognized was the four-star ball necklace. This reminds Roshi of how he, Goku, and Bulma met. Bulma took great offense to that and bonked Roshi.
The part with Raditz telling Goku about his Saiyan origins would be the same. However, when Raditz grabbed Goten. The young boy was kicking and screaming at him. Once the two landed in the wastelands. Goten tries to beat up Raditz while crying. The Saiyan was fed up and placed Goten in his space pod.
The fight with Goku, Piccolo, and Raditz would be the same. Except, Goten did not attack Raditz since he doesn't have any rage powers. Luckily, Goku managed to grab Raditz from behind. As he sacrificed himself when Piccolo used the special beam canon. Goten was shocked and angered to see his dad die.
Bulma, Krillin, and Roshi came over to see what was going on. As Bulma rescued Goten from the space pod. Goten came out to Piccolo and yelled at him. Piccolo snaps and says that he doesn't have a choice. He also mentions the Saiyans' arrival. Goten immediately asked if he could train for the battle. Krillin isn't sure that Chichi would let him. So, Goten stomped and threw a tantrum by saying that he wanted to fight the Saiyans. Bulma looked at Goten and thought of him having Goku's looks and his love for fighting but also Chichi's bratty behavior. Roshi sighs and says they will ask Chichi's permission.