Imprisoned inside Pilaf's palace, the original Dragon Team gets left to die overnight, in danger of being cooked alive as the sun rises above them. Fortunately, it a full moon appears that night, and Goku just happens to have a tail that lets him transform into a great ape.
Oozaru Goku breaks everyone out of Pilaf’s prison, but his rampage knows no limits. With no real sense of self, Goku destroys everything around him, endangering his friends in the process. Working together with Puar, Yamcha cuts off Goku’s tail and saves the day.
Following his death at the start of the Saiyan arc, Goku trains with Kaio in the afterlife. Kaio notably teaches Goku two techniques: the Genki Dama and the Kaioken. The Genki Dama is shown in practice beforehand, but Toriyama saves Kaioken’s reveal for the fight against the Saiyans.
It's worth pointing out that while Kaioken is essentially a technique, Goku comes to use it like a transformation by the Namek arc. Dragon Ball Super goes one further by allowing Goku to augment his transformations with Kaioken, resulting in a new Goku transformation.
Unlike his fight against Vegeta, where he allows Kaioken to destroy his body, Goku bows out before the fight with Frieza gives him the chance. He instead puts all his energy into a Genki Dama, but that only serves to anger Frieza. Frieza retaliates by killing Krillin, but this ends up being his downfall. Watching his best friend burst to shreds right before his eyes, Goku transforms into the legendary Super Saiyan, gains a second wind, and forces Frieza into one of the most humiliating defeats in the series. Super Saiyan remains the most iconic of Goku transformations due to its epic debut.
The Cell arc places a heavy focus on the Super Saiyan state and how to push it further. Vegeta, Trunks, and Goku all have their own augmentations (called Grade forms,) but they’re ostensibly variations of the regular Super Saiyan. It isn’t until Gohan triggers Super Saiyan 2 during the Cell Games that a new proper form is introduced.
Following Goku’s death during the Cell Games, he triggers Super Saiyan 2 while training in the afterlife. It’s likely Goku taps into Super Saiyan 2 sooner rather than later (evident by a lack of stamina drain with a deceased body.) Goku first shows off the form in a flash instance against Yakon.
It’s interesting how Toriyama introduces Super Saiyan 2 for Goku, only to do away with it shortly after. Goku fights Majin Vegeta in the form, but he never fights in it again (in the original manga, at least). Triggering Super Saiyan 3 for the first time while stalling against Buu, Goku's transformation shakes the actual cosmos.
Taking into account the incredibly heavy toll Super Saiyan 3 puts on a living body — and how much effort Goku puts into transforming while dead — it's no wonder the form takes Goku significant time to master. He’s since gotten a firm hold of Super Saiyan 3 in Dragon Ball Super, but its depiction in the Buu arc proves too unwieldy for Goku.
The Super Saiyan 4 transformation gained immense popularity when it debuted in Dragon Ball GT. While this Goku form, and the series it debuted in, weren't canon to Dragon Ball, it still made frequent appearances in video games and merchandise. After being defeated by Baby as a Super Saiyan 3, Goku pursued a greater power.
Goku obtained Super Saiyan 4 by first regaining his Saiyan tail with help from Old Kai, Sugoro, and Shusugoro. This allowed Goku to transform into a Golden Oozaru. Goku initially had little control over this form. When Pan got through to him, Goku gained control of his Golden Oozaru form. In doing so, Goku completed his ascent to Super Saiyan 4, which included a grown-up body.
At its core, Battle of Gods is a character study of Son Goku and the themes that define Dragon Ball. The story centers itself on an unbeatable foe, forcing Goku past his limits but without giving him the actual satisfaction of doing so. Battle of Gods humbles Goku, especially through the act of the Super Saiyan God ritual. Super Saiyan God isn't a form Goku earns on his own accord, but he receives it by the power of others. It’s not something he wants, but he understands it’s needed. At the same time, Beerus recognizes Goku’s skill remains his own even if his power isn’t. There’s a lot to analyze about Goku’s characterization in BoG.
Following the events of Battle of Gods, Vegeta and Goku train with Whis. The anime and manga never reveal how either martial artist triggers Super Saiyan Blue, but everything indicates that they master Super Saiyan Blue during this training period. As Whis is training them in the art of God Ki, it’s likely SSB remains one of their specific goals.
Super Saiyan Blue has since become the "Super Saiyan" of Dragon Ball Super, serving as Goku and Vegeta’s base forms. While the blue pairs quite well with Goku’s Resurrection F attire, it’s not that exciting in his Turtle School uniform.
After Golden Frieza's defeat, both Goku and Vegeta sought to improve upon their newfound Super Saiyan Blue abilities. Vegeta accomplished this later on in Dragon Ball Super, but Goku quickly found a new avenue through the use of an old technique.
Thanks to having greater ki control through Super Saiyan Blue, Goku could incorporate the Kaioken technique. Goku initially planned to save this form for a rematch against Beerus, but felt needed to use its ability multipliers to get past Hit's time-skip technique. Super Saiyan Blue Kaioken had the same drawbacks as the base Kaioken, as Goku's body couldn't handle it indefinitely.
The Universe Survival arc became a major turning point for Dragon Ball Super and basically re-contextualized modern Dragon Ball. All the pieces had been rearranged, the multiverse has been properly expanded, and Goku had a new end goal in mind: mastering Ultra Instinct.
Goku triggered Ultra Instinct for the first time during the Tournament of Power. In the anime, Ultra Instinct was fueled by his own Genki Dama’s energy, whereas the manga had Goku tapping into Ultra Instinct as a mental state. Ultra Instinct was treated like a technique through the series’ actual text, but it’s depicted as a transformation. As of the Moro arc, Goku could trigger Ultra Instinct -Sign- at will so long as his mind was calm. During his fight against Gas in the Granolah arc, Goku opted to use Ultra Instinct -Sign- over the completed Ultra Instinct because the -Sign- variant allowed him to utilize his emotions, such as the desire to win.
Ultra Instinct -Sign- wasn't enough to surpass Jiren in the Tournament of Power. Goku needed to dig deeper by fully completing the Ultra Instinct form. Pushed to the brink of elimination in his Ultra Instinct -Sign- form, Goku remained on the defensive during an onslaught of Jiren punches. After a while, Goku pushed back. This surge in offense culminated in Goku reaching the completed Ultra Instinct.
Much like the -Sign- variant, Goku initially couldn't reach Mastered Ultra Instinct at will. It wasn't until training with Merus during the Moro arc that he reached Ultra Instinct -Sign- again. Goku later triggered the completed Ultra Instinct after Merus' erasure, when Goku kept his emotions in check during a heartbreaking moment. Since then, Goku has used the form at will.