Izuku would likely become more cynical. Stain's views are quite extreme, and being trained by him would expose Izuku to those ideas. He might start to question the motives of other heroes more often.
His fighting style would change drastically. Stain is a very different kind of fighter compared to what Izuku has learned so far. Izuku would become more stealthy and efficient in combat. He might also develop a more ruthless streak when it comes to taking down villains. However, at his core, Izuku's kind heart would still remain, but he would be more conflicted between the traditional hero values and the new ones he's learning from Stain. He could end up being a hero who is more of an outlier, not fully fitting into the typical hero mold but still striving to do good in his own way. And this would make his relationships with his classmates and other heroes more complex as they might not fully understand his new perspective.
In the fanfiction, perhaps Stain trains Izuku because he believes Izuku has the purity of heart to use power in a way that Stain deems correct. Stain could teach Izuku how to analyze opponents better, which would enhance Izuku's ability to strategize in battles. Izuku might then start to question the current hero ranking system during his training. He could grow to be a hero who is more focused on the essence of heroism rather than just fame and recognition, which would make him stand out among his peers and also cause some friction with the more traditional heroes and hero agencies.
Izuku with Star Platinum's power would become more strategic. Star Platinum has unique abilities like time - stop. Izuku would need to learn how and when to use these powers effectively. This would force him to think several steps ahead in a fight, changing his character from someone who might have been more impulsive in combat to a more calculated fighter.
Izuku might become more self - conscious about her appearance in this fanfiction. Since she is a girl, she could face different beauty standards, which may influence her confidence in different ways compared to when she was male - presenting in the original story.
One good 'Stain Trains Izuku' fanfiction could be 'The Training of a Hero' where Stain imparts unique combat skills to Izuku. It shows how Izuku's determination combines with Stain's unorthodox training methods. Another might be 'Stain's Influence on Izuku' which delves into the psychological impact of Stain on Izuku's view of heroism.
The Gamer Quirk also changes Izuku's relationship with others. He can now offer unique insights based on his quirk. For instance, he might be able to analyze a villain's attack pattern like it was a boss fight in a game. His friends and allies can then use this information to their advantage. This makes him a valuable member of any team. At the same time, it also means that he might be seen as a bit of an oddball because his thinking is so different from the norm. But overall, it helps him form deeper connections with those who understand and appreciate his quirk.
You can try looking on fanfiction platforms like Archive of Our Own (AO3). It has a large collection of My Hero Academia fanfictions, and there's a good chance you'll find 'Stain Trains Izuku' ones there. Another option is FanFiction.net, which also has a wide range of fan - created stories in this genre.
He'd probably become more adaptable. Naruto is already flexible in combat, but RWBY's diverse combat styles would force him to be even more so. His personality might also become more open - minded as he experiences the different cultures and ways of thinking in the RWBY world.
Naruto would likely become more secretive. Since Orochimaru's training methods are often not in line with Konoha's values, Naruto would have to hide what he's learning. He might also become more distrustful of others as Orochimaru could instill in him the idea that everyone is out to get him for his newfound power.
If Naruto is trained by Deathstroke in fanfiction, his character would change in multiple ways. For one, he might lose some of his naivete. Deathstroke's world is a harsh one, and Naruto would have to adapt. He could become more calculating and less likely to jump into a fight without thinking. His relationships with his friends might also change. He might start to keep them at a bit of a distance, not wanting to put them in danger as a result of his new, more dangerous fighting style.
Naruto's character would become more serious. Sephiroth's training is no joke and Naruto would have to focus more. He might lose some of his naivete and become more worldly.