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 Frieza
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 recieved 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.