Despite having awful tan lines, Kiyone gets plenty of cheers from the crowd when she's on display for a beauty contest. For a character that has to put up with Mihoshi and is kind of a Chew Toy, it's nice to see her get something good. There are even tears in her eyes.
Ayeka and Sasami in Episode 9. No matter Sasami's 'secret', her sister loves her.
Ryoko holding back in her fight with Kagato because she didn't want to accidentally hurt Ayeka, and Ayeka using her power to protect Ryoko for a moment. It goes to show that despite how much they fight and get on each others nerves deep down they are friends.
Episode 3 has the recently-born Ryo-Ohki's attempts to cheer up Ayeka when she's homesick for Jurai. Especially so after Ryo-Ohki is Ryoko's spaceship reborn, and initially had little reason to like her. Only when the cabbit gave her a flower when she was upset did Ayeka start to warm to her. Also, her burgeoning affection for Tenchi: The two didn't meet on favorable terms initially, especially after a misunderstanding when Ryoko was intentionally yanking her chain by telling her that Ryo-Ohki's egg was actually her and Tenchi's love child. Though when Ayeka sprains her leg, Tenchi helps her find out what became of her missing half-brother and fiancee, Yosho. Convinced he's still alive, Ayeka apologizes to Tenchi for how she acted earlier and returns Yosho's sword to him - especially important since she held it since her introduction episode in the hopes of finding him, and giving it back to Tenchi shows the trust she now has of him.
The final scene between Tenchi and Ryoko, in "Tenchi Forever", mirrors the one between his parents at the end of the first film ("Tench in Love") - from the location to the dialog, and ends with a sepia toned still frame of them together (from 2:10-3:43).
The final scene in Tenchi Universe, in which Ryoko is revealed to have survived and that she is now back on Earth to claim Tenchi for herself.