In one of the team's earliest missions, if not their first, they go up against The Juggernaut. The man who has singlehandedly handed entire teams of more powerful and experienced heroes their collective asses. The New Warriors actually manage to not die until The Mighty Thor stumbles onto the scene and saves them. This may not sound like much, but in context, it's quite an impressive feat.
The entire battle against Juggernaut is one for Juggy himself. No matter what anyone throws at him, he can take it. When Night Thrasher uses his most lethal weapons, Juggernaut retorts they don't even register on him. When Firestar blasts him with her heat rays until his armor is glowing white hot, he doesn't even feel it. Nova and Namorita sandwich him between two train cargo carriages, and he literally shrugs them aside. Thor blasts him with a lightning bolt, and is visibly gobsmacked when Juggernaut sneeringly informs him that it "tickled". Then Thor conjures up a mystic blast, and Juggernaut actually manages to push back against it. It's only the ground giving way under Juggernaut's feet under Thor's mystic blast that allows the group to finally defeat him; Nova and Namorita bury him in steel girders, which Firestar melts into a cocoon around him, giving Thor enough time to teleport Juggernaut to some alien world. That's the only reason they all get out of the fight with their skins intact.
Vance refusing to be sprung from jail.
This requires some elaboration. Vance grew up in an abusive home, with his mother and himself being smacked around by his father due to Vance's being a mutant (and as it would be revealed, Vance's dad's frustration at repressing his own homosexuality). After having his arm broken by one of Galactus' heralds and being tortured by the mutant immortal Gideon, Vance came home to having his father start whaling on him. Vance telekinetically put his dad through a wall, resulting in his death. He turned himself in and while he was acquitted of murder, he was convicted of manslaughter. Firestar convinced Namorita and Nova to spring him from the convoy taking him to jail, but Vance steadfastly refused. He told his teammates that he had been fairly tried and convicted and would serve his sentence, saying that just because he had the power to free himself, didn't give him the right. Incorruptible Pure Pureness, thy name is Vance Astrovik.
Vance verbally schooling Cap and Iron Man, which also serves as a not so subtle Take That! at the way superhero teams are marketed, as well as the insistence the New Warriors couldn't be relaunched again due to Stamford.
Iron Man: What made you think this (reforming The New Warriors) was a good idea?
Speedball: The part where we saved the world?
Iron Man: The New Warriors are a tarnished brand.
Justice: Did you just call us a "brand"?
Captain America: You and Speedball are welcome at the tower, Justice, as Avengers. You know this. But we can't ignore what happened at Stamford, Vance. There's never been a tragedy like that before and a lot of people still blame the New Warriors for it. I think it'd be better if your team was folded into the Avengers.
Justice: Where you can watch us, is that it? Bottom line: After everything we've done before and after Stamford.. you still don't trust us.