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. And hey, the very last line of the novel is Conan asking his defeated enemy to find Zenobia for him, because he wants to marry her!
While on the run from soldiers who are tracking him with magical aid, 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?
He fiercely hugged the head of the great dog growling at his side, muttered: “All right, boy, give 'em hell!” and sprang to his feet, drawing his ax. Then the dark figures flooded over the breastworks and closed in a storm of flailing axes, stabbing knives and ripping fangs.
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 modern fantasy heroes, and is the origin of "sword and sorcery" and the so-called "low" Heroic Fantasy. Dungeons & Dragons owes half its existence to The Lord of the Rings. The other half it owes to Conan.
Conan's return to Aquilonia after he escapes capture in "The Scarlet Citadel". He arrives on the back of a magically conjured beast, lifts the attempted usurper over his head by throat and crotch before throwing him off a balcony ("Take your plots to hell with you!"), then calling all his loyalists together 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.
Another awesome moment from The Scarlet Citadel is when Conan give an epic "The Reason You Suck" Speech to the king of Ophir.
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!
Making things even more impressive is that, after Baal-pteor falls dead, Conan privately admits that he was bluffing; a few seconds either way, and Baal-pteor would have killed him instead, an honor that very few human foes can claim in the Conan mythos.
And killing the demonic Dark Man with a thrown solid silver bench in The Black Stranger. SILVER AND FIRE!
Subverting the Friend-or-Idol Decision in The Jewels of Gwahlur when he doesn't even hesitate for a second to save the Girl Of The Novel instead of the titular (and priceless) treasure.
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.
When you're hanging from a cross and kill the birds that have come to peck out your eyes... with your teeth... you have entered the very heights of Rated M for Manly.
Valerius, the second main character of the story, being in charge to rescue the queen while Conan storms the gate. After spending most of the story mending his wounds and passing as a beggar to gather info he springs into action and sink his sword through Salome's heart.
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 death of her brother Bhunda Chand, that leads off the book, is somewhere between this and a Tear Jerker. The great prince, loved by all the people, is dying of an evil curse that drives him insane. He suddenly becomes lucid and commands Yasmina to kill him while he is in his right mind. And she does.
"Swiftly, I command you!" There was the old imperious note in his failing whisper. "You have never disobeyed me — obey my last command! Send my soul clean to Asura! Haste, lest you damn me to spend eternity as a filthy gaunt of darkness. Strike, I command you! Strike!"
Yasmina spends the story (mostly) unwillingly adventuring next to Conan. In the end, she gets exactly what she wants, though definitely not in the way she wanted — Conan has killed the Black Seers of Yimsha. Despite extremely high sexual tensions, she prepares to return to her country and throne with her army, though she offers Conan a ransom for services rendered. Conan, leading the barbarian tribes arrayed against Vendhya, says he'll come collect his ransom — from Yasmina's capitol, with fifty thousand men to ensure the scales are fair. Yasmina laughs in his face and promises to meet him with a hundred thousand men. The two part considering each other a Worthy Opponent (and remember Yasmina has seen Conan in action extensively, and is under no illusions about how formidable an opponent he really is). Perhaps the only character in the Conan canon with balls bigger than Conan's is Yasmina.
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.
Films
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."
