Forward is packed with these, especially in the first story arc. One of the more potent ones is when Jayne and River are being tortured by Niska, and he manages to grab her hand and hold it to keep her from breaking. Or any scene between them while they are escaping Niska's ship, with Jayne forced to carry her on his back. Later on in the 3rd arc, where Mal tries to comfort River after she has a mental breakdown after killing Ott's pirates.
"Say again?" he asked. "Home," she said, and she started crying into his neck, her body shivering with sobs. "I want to go home, Jayne." "Me too," he hissed, and started out the door with the featherweight of little crazy person on his back. Pistol leading, Jayne Cobb stepped out of that hell chamber, and started their escape. He was gonna get her out of here, safe and sound, even if the whole gorram Alliance stood in his way.
The whole of the "Adrift" arc's epilogue, especially the bits between Book and River.
The "First Interlude" chapter, which involves River making a crucial, personal decision on her eighteenth birthday: to stand on her own.
She wouldn't despair anymore. She wouldn't be helpless. She was a person - damaged, frail, and insane, certainly, but a person nonetheless. No longer a child, no longer a plaything, no longer a burden to be borne on a shoulder or a charge to be safeguarded or a weapon to be feared.
The epilogue for Condor.
The epilogue for "Silver" has an adorable scene between Kaylee and Simon, and also has a powerful scene where River confesses to Book what she feels was a very bad thing she did, with flashbacks to her confronting one of the men who hurt her at the Academy who himself deeply regrets what he did. Ultimately, he asks for her to forgive him, and River does.
The ending to "Fourth Interlude." River has been put through so much over the last few chapters, what with the Mind Rape, losing control of herself and flying into berserk frenzies, then crashing on a border planet and forced to walk across a desert on serious injuries, all the while suffering hallucinations and then getting horribly injured fighting a group of pirates. At the end of it all, after she passes out, River has a flashback to when Simon rescued her from the Academy, and then comes to. The reader's probably expecting something even worse for River when she wakes because, well, that's how the story's been going. Instead, she finds herself in a hospital, being tended to by an old doctor who she can tell is honestly concerned for her safety, and she finally lets herself relax and feel safe. After everything else she's been through recently, encountering a decent human being on the Border who isn't scared of her and just wants to help her recover is a powerful scene in its own right.
The epilogue for "Last Man" has all the scenes with Wash and Jayne. Both men have been struggling in life-or-death situations in the last story, and Jayne is battered and wounded, and has to lean on Wash the whole way. And when Jayne tells Wash that he owes him for saving his life, Wash refuses, because they're crew. And Jayne...doesn't know how to respond.