Planning to take the slave girl Zenobia with him after she rescues him. Even though he couldn't, the fact that he wanted to says a lot about his honor.
Being on the run from soldiers (who are able to follow him because of a bewitched crow that keeps squawking over his position), he sees an old woman in the borders of the kingdom he lost about to be lynched. Forgetting all about his problems, he goes to protect his subject.
Captured and enslaved while pursuing the Macguffin, he not only breaks free, but whips the black slaves into a frenzy when they realize he's the man who led scores of their people as pirates against those who'd enslave them.
The scene where Conan, disguised as a mercenary, contemplates abandoning his quest and returning to the exciting and boundless life of a wandering adventurer. However, his honor and sense of duty to his subjects prevents him from leaving his kingdom in the hands of a tyrant.
In "The Queen of The Black Coast", after losing the first true love of his life, Conan waits for the demonic winged ape that hung her by her necklace. With a bow, his sword, and his bare hands, he fights off twenty were-hyenas that serve the creature.
Belit, the aforementioned true love, coming back from the dead as she promised to give Conan a fighting chance against her murderer.
In "Xuthal of The Dusk", also known as "The Slithering Shadow", Conan comes face-to-face with an Eldritch Abomination. At first, he wants to flee. He then sees that it's frisking his love interest of the story with its tentacles. Cue awesomeness as Conan precedes to tear into the beast with his saber and poniard. Did You Just Punch Out Cthulhu? much?
Conan himself in general, as he's continued to be a part of pop culture and literature seventy five years after his last original story, may be one of the first fantasy heroes, and is the origin of Heroic Fantasy and Sword And Sorcery.
Conan's return to Aquilonia after he escapes capture in 'The Scarlet Citadel'. He arrives on this mystical bird thing, grabs the supposed "rightful heir" and chucks him into a crowd of displeased citizens, right before rounding up an army to kick the crap out of the other invaders. You can just hear the song "Holding Out For A Hero" being sung in your head while this is going on.
Conan: I found Aquilonia in the grip of a pig like you—one who traced his genealogy for a thousand years. The land was torn with the wars of the barons, and the people cried out under oppression and taxation. Today no Aquilonian noble dares maltreat the humblest of my subjects, and the taxes of the people are lighter than anywhere else in the world. What of you? Your brother, Amalrus, holds the eastern half of your kingdom, and defies you. And you, Strabonus, your soldiers are even now besieging castles of a dozen or more rebellious barons. The people of both your kingdoms are crushed into the earth by tyrannous taxes and levies. And you would loot mine—ha! Free my hands and I'll varnish this floor with your brains!
Conan: I think you never saw a man from the West before. Did you deem yourself strong, because you were able to twist the heads off civilized folk, poor weaklings with muscles like rotten string? Hell! Break the neck of a wild Cimmerian bull before you call yourself strong. I did that, before I was a full-grown man — like this!
Surviving a crucifixion at the start of A Witch Shall Be Born...and then visiting the same fate upon the bastard who crucified him in the end.
Yasmina Devi, heroine of The People of the Black Circle refusing to break under the villain's Mind Rape, then convincing Conan to help her fight off an invasion of her country, and personally taking charge of her army to direct the fight.
The Phoenix on the Sword when assassins burst into King Conan's bedchamber intending to slaughter an old man in his sleep, or at worst while he is still waking up. On finding themselves face to face with a half armored savage, they stop dead.
From the 1982 film, Conan's prayer to his god Crom, which doubles as a Rage Against the Heavens. Essentially, he's saying "I'd love it if you'd help me kick these guys' asses, but if you won't, then go get bent."