Goku's understanding of different universes and races could be deepened in fanfiction. Since Dragon Ball Super introduced the multiverse concept, fanfiction could explore how Goku interacts with and learns from beings from other universes, making him more worldly and less naive in some ways.
Super Saiyan 2 was a transformation form of the Saiyan in the Japanese anime Dragon Ball. According to Goku, it was "a Super Saiyan that surpassed the Super Saiyan." Goku had shown this form when he fought Vegeta in the Buu chapter. His strength could easily defeat the complete form of Sarroux. Before the Shalu Chapter, Goku had already reached the limit of his Super Saiyan state when he came out of the mental time room. When Sarroux was playing, Goku's Super Saiyan was at full power, and his energy reserves were fixed. At that time, Gohan's transformation (not explicitly Super Saiyan II and not necessarily a traditional Super Saiyan) reached the state of lightning. That was Gohan's exclusive transformation mode when he broke through the limit of anger and potential. Not every Saiyan could achieve it. If there was no follow-up Buu chapter, Lightning Gohan might be the ultimate form of Gohan's full potential. However, in the Buu chapter, Goku displayed his Super Saiyan 2 form when fighting Vegeta. The shapeshifters who had transformed into Super Saiyan 2 included Sun Gohan, Sun Wukong, Vegeta, Future Trakus, Kaliphra, Gaga, Kaiphra, etc. This form appeared many times in Dragon Ball Z, Dragon Ball MT, Dragon Ball Super comics and animations, and Dragon Ball Heroes, and derived many enhanced forms. The combat power of this state was twice the full power of the Super Saiyan, which was 100 times the normal state (from the Super Excitement Guide: Story). In essence, it was twice the power of the Super One in a short period of time. The total energy did not change, so the physical consumption was twice that of the Super One (according to Toriyama Akira). If he trained hard, he would be able to unleash the power of a Super Saiyan 3 (see "Trakus"). You must become a Super Saiyan or a Super Saiyan at full power before transforming.
One possible fanfiction idea could be that Goku's daughter has unique powers that are a combination of his Saiyan strength and something new, like the ability to control time on a small scale. She might struggle to fit in with the other Z - fighters at first because they see her as just Goku's daughter, but she proves herself by facing a new, powerful enemy that the others can't defeat easily.
One of the interesting power - ups could be his Ultra Instinct form. In fanfiction, it might be further enhanced, like having different levels that are not shown in the original. For example, he could have an Ultra Instinct - Kaioken combination which would be extremely powerful.
Caulifla and Goku's relationship in fanfiction could be really interesting. Maybe in this fanfiction, they start as rivals. Caulifla, with her Saiyan pride, wants to prove she's stronger than Goku. But as they fight more, they begin to understand each other's fighting styles and develop a mutual respect. This could lead to some really cool training sessions where they both learn new techniques from one another.
In this fanfiction scenario, it could be an extremely emotional encounter. Goku, who has grown so much in Dragon Ball Super, finally meeting his father Bardock. Maybe Bardock gets to see how his son has not only become a powerful Saiyan but also a protector of Earth. It could lead to some interesting exchanges about Saiyan heritage and the future of the Saiyan race.
They could start by sharing more personal stories. For example, Goku could tell Bulma more about his time on Planet Yardrat. This would deepen their understanding of each other.
In many Dragon Ball Super fanfictions, Gohan's character development can take various directions. Some fanfictions might focus on his return to his Saiyan warrior roots. For example, he could start training intensively again after realizing the threat to his family and the universe. This would involve him pushing his limits in combat and rediscovering his latent powers.
Her character could be developed by giving her unique fighting styles. For example, she could combine Goku's martial arts with some new techniques she invents. This would show her creativity and independence.
To develop Goku Jr's character further, he could be given more complex emotions. For example, showing his fear in the face of truly overwhelming enemies, rather than always being fearless like Goku. This would make him more relatable. Also, exploring his relationships in more depth, like his friendship with Pan. Maybe they have a falling out over different views on how to protect their world, which forces Goku Jr. to grow and understand different perspectives.
In the Dragon Ball Super manga, Goku's main concern is his battles and training. There's no real 'affair' in the traditional sense. His story is more about his quest for power and defending his friends and the Earth.