It's really hard to say. Goku has his Super Saiyan forms which give him incredible power boosts. But Superman has super strength, heat vision, and is nearly invulnerable. In a short fight, Superman might have the edge due to his instant powers like flight and super strength without having to power up. But if Goku manages to reach his higher Super Saiyan forms and uses techniques like the Spirit Bomb, he could potentially defeat Superman.
One idea could be a battle between Goku and Superman. They could start off as rivals, each thinking their own planet needs more protection. As they fight, they gradually realize they have more in common than they thought, like their strong sense of justice. Then they team up to face a common enemy, like Darkseid and Frieza working together.
Goku would be really excited to meet all the different heroes in the DC universe. He'd probably want to spar with everyone right away. Flash would be intrigued by Goku's ability to move at high speeds using his Ki. They could have a friendly race or competition to see who's faster in different situations. At the same time, Piccolo could team up with Martian Manhunter. They both have the ability to sense things from a distance and could use their skills to detect threats to both universes.
Maybe a story where the Dragon Balls are accidentally sent to the DC universe. This would attract the attention of characters like Batman who would be intrigued by their power. Then, characters from Dragon Ball would have to travel there to retrieve them, leading to all sorts of encounters and battles.
In some fanfictions, Vegeta might team up with Batman. Vegeta's combat skills and Batman's strategic mind would make for a great combination. They could work together to take on a powerful villain that threatens both the Dragon Ball Z and DC Universe. For example, a villain who has the ability to steal energy like Cell and the mind - controlling powers of Brainiac.
Another possibility is that 'American Dragon' could be a name used for a product, a brand, or a local mascot in the United States. For instance, a sports team might choose 'American Dragon' as their mascot to represent strength and fierceness, similar to the characteristics associated with dragons in mythology.