Tris taking in an orphaned Glaki, planing on bringing her to Rosethorn and Lark when they get back home.
For that matter, Tris taking in Chime, a strange magical creation after protecting it from its frightened creator.
Who she later genuinely apologizes to and protects vehemently from potentially being tortured to give the local legal authority the answers they want.
Keth giving Tris a necklace made from Chimes flames, then telling her that he has no intention of taking her back.
Keth: I think she (Chime) belongs with you.
Briar's reasoning behind the date he chose for his birthday at the end of Briar's Book/The Healing In The Vine. "The girls would be vexed at having so little time to prepare, but they would adjust. They would know, as he did, that his life began when Rosethorn had invited him into her world."
And before that, when Rosethorn was dying, she said to Daja "My boy. You three girls—look after Briar. When I'm gone." She doesn't stay dead, but it doubles as a Tear Jerker and a moment of heartwarming.
The last lines of the last book. Briar explains the little mind palace created for the four in his dreams.
Briar: It's us.
All: It's always us.
In The Will Of The Empress, when Sandry first gets kidnapped, Briar and Tris, despite not getting along with her so well, instantly drop everything and go to help as soon as they find out what's happened.
"He is a faithful son," the First Dedicate said [...]. "You have been unstinting with your own love, that he is so unstinting with his love for you."
Briar and Evvy reuniting after Briar thought she was dead, complete with Cry into Chest.
Daja, Briar and Tris all working together to make a magical light-giving crystal to help Sandry with her fear of the dark.
Just the whole story of the main four characters. When they started Tris and Briar hated everyone, Daja only liked Sandry- who was the only one trying to be at all friendly. They grow into the greatest friends ever, even calling each other siblings.
Tris offering to teach Briar how to read and him accepting, keeping in mind they spent the whole book angry and hating pretty much everyone.
It's a tiny moment, but in Cold Fire, a messenger gives Daja a message while she's working with a smith, at lunch time. The messenger, who works for the Ladraduns, is hungry (his boss doesn't feed her workers), and so the smith Daja's working with gives him some food and tells Daja to read the message carefully. Daja, getting his idea, keeps rereading the message until the messenger has finished his food and is given more. Then she gives him a huge tip.