Yukari-Sensei telling Kagura in the last episode that she does not need a memento from her time at high school, as she has all her fond memories in her head. Cue Kagura bursting into tears. Doubles as a Tear Jerker.
Mayaa coming back to save Sakaki and Chiyo from Kamineko and the local cats.
Not to mention, Sakaki protecting Chiyo. Even though she's scared too, she's staying strong and protecting her classmate.
Sakaki staying at Chiyo's place for the night is one big Heartwarming Moment, since we get to see a side of Sakaki we haven't really seen before.
There's also the part in the anime in which Chiyo tells Sakaki that she looked cute playing with Mayaa. Considering Sakaki's issues with how she appears more intimidating than cute, this means a lot to her.
And listen to how Sakaki responded to being called cute; she blushed and said "You...you think so?" Not "You...saw that?", as though she were ashamed that someone witnessed her acting in opposite to her supposed 'tough' demeanor, but "You think so?", as though she finally found somebody she could open her heart to and really be herself. It's in that moment when it's all but spoken that Sakaki and Chiyo-chan pretty much see each other as the younger/older sister they wished they had.
The Blam of episode 09, where the girls are running and laughing among the clouds, particularly Chiyo and Sakaki lying on top of Tadakichi-san, laughing and hugging. It was probably just in Sakaki's mind, though. Still, it shows how much she loves her friends.
The end of episode 12 has Sakaki and Chiyo playing jump rope, after Chiyo had earlier reminisced about playing it with her primary school friends. Then Tomo shows up...
What makes it really heartwarming is the way it starts: After watching some girls playing jump rope in the park, then leaving it behind when they go back home, Chiyo and Sakaki walk over. Sakaki then says "I know we're still walking but...*looks at Chiyo and smiles*...you wanna play for a bit?" This is the first time we see Sakaki showing her gentle side to anybody, and the fact she chose to show it to Chiyo-chan is proof that she loves her very much, almost like a Cool Big Sis.
With Yomi and Osaka, all of whom join in, the whole thing cements into the viewer's mind just what kind of girls these are and the friendship they all share, and how much they care about Chiyo.
Before that, she got to pet Tadakichi-san, and was so happy about it that she petted him for hours. It was both hilarious and heartwarming.
Chiyo, while in the penguin outfit, has trouble eating stuff because she can't get the food to her mouth while in the penguin suit. Osaka helps her out, and Chiyo reacts with sheer glee. It is the cutest. Thing. Ever.
Kagura and Chiyo hanging out in episode 19.
In the second last episode, Osaka makes up her own good luck charm to help pass her entrance exams: splitting chopsticks evenly. Everyone seems to make fun of her at the time. Later on, when Osaka goes for her final exam, we see Chiyo sitting down with a huge pile of single use chopsticks, splitting them in half, all for the sake of her friends.
Tomo starting a standing ovation for Chiyo in the last episode.
To elaborate, this is Tomo, who throughout the course of the series has been known to be the meanest and least sensitive of the group (it's her way of showing affection), she's liked to tease Chiyo up until this point in a mean, older sister, "because I can", kind of way. Then she seems to be the one who is the happiest for Chiyo at that moment and the most proud too. If anything, it's the moment that tells us she really does care.
Yomi running to and thanking her friends when she finds out she has passed her entrance exam.
The six main girls bowing to Yukari-sensei and saying "Thank you very much". Although the viewer doesn't hear it, the text comes up on screen, implying they did.
The final scene of the anime, where Chiyo says to herself that even though school has ended, the girls will still be the best of friends forever.
Sakaki rolling on the floor in happiness with Mayaa.
In the recent supplementary lessons, Osaka makes a stamp out of her eraser, and says she will stamp the book of anyone who is nice to her. She borrows Yomi's notebook, and before Yomi can finish saying she doesn't need any stamps, we see that Osaka has stamped 6 times in Yomi's book, and written "Thanks!" in between them. Awwwwww.
The second Athletics Day episode, with Kaorin running alongside Sakaki.