Bakugo and Midoriya are the main characters in My Hero Academia and are the series rivals and probably the strongest students at U.A. (other than the big three). They have been butting heads since they were 4 years old, Bakugo always trying to be better than Midoriya, and Midoriya just always trying to be the best he can be.
These two characters are definitely destined to be the best pro heroes in Japan, and though the series narration makes it apparent that Deku will be the number one hero, Bakugo surely cannot be far behind and may even take up the mantel time and time again himself.
Bakugo: Recognizing Others
While Bakugo is an extremely hot-headed individual with an ego stronger than a lion, he occasionally has moments where he takes a second to acknowledge others' strength compared to his own. He acknowledges that both Eijiro Kirishima (his closest friend) and Ochako Uraraka were going to be a challenge for him. With Uraraka, he fought her during the tournament arc, and though she was terrified to battle him, he actually gave her some encouragement saying that he acknowledged her strength. He also acknowledges Midoriya's strength and that it is growing, which is why he wants to battle him all the time.
Midoriya: His Desire To Help Others
One of the main reasons Midoriya really wanted to become a hero, other than his obsession with All Might, was due to the fact that he really wanted to help others in their greatest time of need. Even when he didn't have a quirk, he still rushed in to help people whenever he could. He ran in against a supervillain who had latched itself onto Bakugo in order to try to save the boy who had been bullying him for years.
Bakugo: Determination To Win
Bakugo is extremely determined to become the best hero there is. He wants to be number 1 and never seems to let anyone get in his way, not even himself. Bakugo is fiercely determined which is what seems to continuously fuel the fire within him. He doesn't seem to care how injured he gets, or how exhausted his powers become, Bakugo continues to perceive.
This train of thought made some of the U.A. teachers and the League of Villains that Bakugo had the perfect making for a villain, however, he again proved them wrong by yelling at them even when he was in danger. He has the mentality of "Winning to Save" which contradicts with Midoriya's.
Midoriya: Saving To Win
Though All Might told he must learn how to "Win to Save" from Bakugo, Midoriya's mindset of "Saving to Win" makes him a pretty great hero. Midoriya doesn't care about winning or looking cool. He doesn't care about being the best. All he cares about is saving people, making sure that they are safe, even if he is not. This is a pretty big deal when Midoriya saves Kota from Muscular. He puts himself in harm's way, which is why he keeps breaking his body in nearly every fight.
Bakugo: Heroes Rising
Bakugo really proved himself in the Heroes Rising film. He and Midoriya needed to team up in order to stop the villains from taking over the island Class 1-A had been working on. Bakugo begrudgingly found himself bonded to two children but continued to take care of them even while he continuously threatened them. Bakugo also allowed Midoriya to give him to the power of One for All to help him in the final fight, even though he broke his bones in the process. Bakugo, with Midoriya, learned how to Win and Save during this film.
Midoriya: Hero Obsession
It is no secret that Midoriya is incredibly intelligent. Not only is he smart about the basic school subjects, but he also is vastly knowledgable about quirks. He is always murmuring off the best ways to stop his opponents and spew out facts about the quirks he sees. In these instances, though they are often comedic, Midoriya is dissecting each character and their quirk, trying to figure it out in a moment's time. He is able to deduct all of their strengths and weakness in order to figure out the perfect strategy on how to combat his enemies.
Bakugo: Intelligence
And though Bakugo definitely seems like the type of character who doesn't care about brains and wit, only brawns and fire, he is vastly intelligent as well. Like Midoriya, Bakugo analyzes the quirks around him and is able to deduct the best method of fighting those he goes up against. He shows this during the sports event while watching Tokoyama. He knew the best way to fight him would be to surround him in pure light, so that his shadow may not help him.
Midoriya: Hero Killer
Midoriya, alongside Todoroki and Iida, fought against the Hero Killer known as Stain. Stain had killed a great number of heroes at this point without being stopped by anyone, up until his fight with the three students from U.A. after he nearly killed Iida's brother.
This was a really big moment for Midoriya's hero career. He had to discover the best way to go about stopping Stain and make sure his friends (and him) stay alive against someone actively trying to kill them. The pro heroes were all too late to the scene and only arrived after the boys were able to incapacitate him.
Bakugo: The League Of Villains
At the end of the summer training camp arc, Bakugo gets kidnapped by the League of Villains, in hopes to make him the next supervillain. However, Bakugo refuses to be manipulated and refuses to be a victim. During the fight between All Might and All for One, Bakugo manages to break free and fights pretty much the rest of the League alone, particularly Twice, Toga, and Mr. Compress. This was an amazing moment for Bakugo just after he shot himself up in the air to grab Kirishima's hand and escape with the rest of the 1-A crew so that All Might may focus on his own fight.
Midoriya: Eri
Midoriya's fight against Overhall was an incredibly big deal. It was the first fight in which All Might could not come to help due to his health condition and left Midoriya to do pretty much all of it on his own. He had help, of course, but when it came to the actual fight against the villain, it was more of a one-on-one. Midoriya was also extremely careful in making sure Eri was safe during the entire battle as she both fueled his power and sucked his energy dry. It was the biggest hero moment for Midoriya thus far.