The opening scene, where Conan's father tells his son of the Riddle of Steel. Becomes more heartwarming in hindsight, due to the obscene brutality that follows. You can imagine Conan clinging to this memory of his father during his time as a slave.
See also Conan's mother shielding her son from Thulsa Doom and his riders.
King Osric's parting words to Conan and his friends.
"There comes a time, thief, when the jewels cease to sparkle. When gold loses its lustre and the throne room becomes a prison. And all that remains is a father's love for his child."
The Director's Cut DVD has an extra scene before the final battle where Conan, knowing that their chances of survival are not good, offers Subotai the chance to leave. Subotai declines the offer to run away and chooses to stand beside his friend.
That's also a nice scene because it shows Conan and Subotai actually being happy and just having a casual conversation, albeit one about the futility of ever finding peace in their lives when they work as thieves and mercenaries.
Subotai: No. It would only lead me back here one day. In even worse company.
Valeria promising Conan that she'll always be there when he needs her, even if doing so requires her to return from the afterlife. Later on she makes good on her promise, appearing to aid Conan in the final battle, and clad as a valkyrie.
Valeria: All the gods they can not sever us. If I were dead and you still fighting for life I'd come back from the darkness - back from the pit of Hell - to fight at your side.
During his time as a pit fighter, Conan is given a woman to have sex with as a reward for his many victories. The woman has been stripped naked, shoved into a room with a hulking barbarian, and is naturally frightened, expecting him to ravish her. But Conan instead gently leads her away from the leering eyes of the onlookers, covers her up, lays her down and looks her over with a gentle eye, wordlessly reassuring her that he won't harm her.
Conan is passed-out drunk, and Valeria tries to filch one of their jewels. Conan suddenly awakens, grabs her wrist, and takes the stone out of her hand. Then he smiles and places a larger one in it. Two scoundrels in love.