Goku

Goku was the first of the saiyans to turn Super Saiyan. This transformation occurred on Planet Namek where he first releases his enormous power after the horrible Freiza kills his best friend, Krillin. He was enraged and goes to the state of a Super Saiyan. He would later achieve Ultimate/Ultra Saiyan in the Hyperbolic Time Chamber later in the series. Goku then reaches the level of Super Saiyan 2 in the afterlife (because Cell killed him) by training hard in the Buu Saga. During the Buu Saga, Goku faces off against Fat Buu. Since he is so strong and Goku can't defeat him he gets very angry. This anger pushes him into Super Saiyan 3. In Dragonball GT, Goku turns into a Golden Oozaru. When he sees his granddaughter (Pan) crying he turns Super Saiyan 4.

Gohan

Gohan is one of the strongest character in the Dragonball Z. He just doesn't have the will to fight like father does. He first achieves the level of Super Saiyan in the Hyperbolic Time Chamber. There, he thinks of all the times he let his friends down, how much they have been hurt because of him. The past memories push him to the next level. He then achieves his next level when fighting against Cell in the Cell Games. Mr. Satan throws the head of Android 16 in front of Gohan so that he can speak to him. Android 16 started to tell him very touching story about how he loves the Earth and tells him to fight. But suddenly, Cell steps on the head and kills Android 16 for good. This enraged Gohan and pushes him to the level of a Super Saiyan 2. This stage is the highest that Gohan can go up to.

Vegeta

 Vegeta first reaches the level of Super Saiyan in space. On Earth, Vegeta goes under strenuous training. He then goes into space and trains, but would get badly injured. The injuries made him to realize that he doesn't care of becoming stronger than Goku anymore. He grew angry and transformed in a Super Saiyan. When Majin Buu made Vegeta a Majin follower, which made him stronger, the power that he received grom that was enough to send him into Super Saiyan 2. Vegeta never reaches the level of Super Saiyan 3, but he does reach Super Saiyan 4. His wife, Bulma, made a device called Brute Wave, which he uses to turn Super Saiyan 4.

Mirai Trunks

Trunks come from a different timeline (the future). When Androids 17 and 18 kill his mentor and best friend, Gohan, he gets very angry and goes Super Saiyan. The farthest Trunks go is Ultimate Saiyan, which he achieves in the Hyperbolic Time Chamber.

Trunks

 The exact time that Trunks goes Super Saiyan is unknown. He does it in the Buu Saga at the age seven or eight. He achieves the level of Super Saiyan While training with his father, Vegeta.

Goten

Goten reaches the level of Super Saiyan at about the same age as Trunks. He reached this stage when fighting against Trunks in a tournament. He did not need to become enraged to become a Super Saiyan.

Gotenks

 Gotenks is a fusion of Goten and Trunks. In the Buu Saga he achieves the level of Super Saiyan 3.

Vegetto

Vegetto is a fusion of Goku and Vegeta, with Vegeta as the dominant character. Since Goku and Vegeta were already Super Saiyans, he did not have to reach the level, he was already there.

Gogeta

Gogeta is a fusion of Goku and Vegeta, with Goku as the dominant character. Since Goku and Vegeta were already Super Saiyans, he did not have to reach the level. When SSJ4 Goku and SSJ4 Vegeta fused, they formed Super Saiyan 4 Gogeta, the most powerful character in the whole Dragonball Manga.