In the manga, the first nice thing Umi has to say to the other two is after noticing their good form when first wielding their weapons. That becomes the starting point for dropping her Rich Bitch persona.
In episode 7 of the anime, Fuu goes to save Ferio all on her own, and succeeds.
Hikaru, Umi, and Fuu explaining why the vision they see in the Spring of Eterna are the ones most precious to them; Hikaru saw her dog, Hikari, Umi saw her parents, and Fuu saw herself. Also counts as a Tear Jerker.
Hikaru: He's one of my best friends! He was always right by my side whenever I had to cry! When I hurt myself he'd be there to comfort me. He can't speak to me in words, but we can understand each other just fine! He's one of my closest friends in the whole world!
Umi: These are my loving parents! They sheltered me with their love every day when i was growing up and I saved it up in my heart all this time! And I'm going to return all of it by taking care of them when they need me and grow old! I haven't had the chance to repay any of their kindness yet; there's no way I'd ever hurt my mother and father!
Fuu: I am extremely important to myself! If something were to happen to me, my father, my mother, my sister... everyone who loves me would feel the pain! I am still young yet, and have done nothing for those who love me, so at least I need to tell them I appreciate their love! That is all I can do at this point. That's why I am important to myself!
Umi's weapon evolves in episode 9 when she breaks through the Lotus-Eater Machine and accepts that she's got to become a Magic Knight. She makes it back just in time to save Hikaru and Fuu from Alcyone.
When Fuu is able to break Caldina's enchantment through sheer willpower, Caldina assumes that the girls must have known each other for a long time to be so devoted. Fuu corrects her.
Caldina: You'd risk your life for a couple chicks you just met? You nuts?!
Fuu: Time is of little consequence to the heart. Even if our time together has been brief... and even if I know only a little bit about them... it only means I have more to look forward to in our friendship. My heart knows this—I want to fight with them.
The eighth episode shows how much of an animal lover Hikaru is.
The scene in the episode "Hikaru, Umi, and Fuu’s Unyielding Wish", when Mokona pushed Hikaru out of the way of an oncoming monster attack.
Umi and Celes during the beginning of the episode "Hikaru and Friendships Torn Apart". Umi discovers that she's being held prisoner. Upon hearing from Celes that he is alive and well, she starts asking him about his past and to speak to her more informally in the future. It's a very adorable moment for the girl and her magical Giant Mecha companion.
Umi: I have to admit, when I first met you I thought you were pretty scary looking. But your so kind. I wish you would tell me something about yourself. How you spent all those long long years. You should tell me your memories.
Celes: Very well, young girl from another world.
Umi: Please, call me Umi. I'm calling you by your name, Celes, so I want you to call me by my name too, I mean if that's alright?
Umi: (smiles brightly) Hey, I appreciate that, Celes. You can be sure I'll call your name right away if I'm ever in any danger.
Near the end of the anime, Lantis finally admits that he loves Hikaru back.
In episode 22 of the anime, The girls see "Presea" again and Hikaru ran to "Presea"'s arms, overjoyed was an understatement to describe how the girls felt to see "Presea" again. Since Presea died in season 1 of the anime when the roof of her work room caved in on her. This could also count as a tearjerker because the girl they believed to be "Presea" was really Presea's younger twin sister, Sierra, and we later learn that she also likes the girls but feels like crap for keeping the charade in front of them, even when she believes it's for the best.
The last scene of the anime, with the trio seeing a glimpse of Cephiro being restored to its former glory.
Rayearth II starts with the girls at home, and their families deeply concerned over their depression since the class trip to Tokyo Tower. Even if there's no way they can know what happened, they care a great deal. Hikaru's eldest brother says that as worried as they are, it's all right if she can't talk about it. Fuu's older sister Kuu says something similar—that she should find someone who will understand and talk about it, even if it's not her or their parents.
Something of a meta example: The end credits for the final episode of the first season play over the image of the entire Earth that was the end result of the zoom out from the girls back in Tokyo Tower and the music is a more melancholy rendition of the usual ending theme. Considering how the episode ends, it's really sweet that there's a tacit acknowledgment that the usual jouncy music and animation for the end credits would result in a massive case of Mood Whiplash.