Episode 2 has Mugi "bargaining" with the music store clerk. Not only does she do it so Yui can afford her guitar soulmate but it's the most adorable insistence of haggling ever seen on screen.
End of episode 9 "New Club Member!"
A lot of episode 12 "Finale K-ON!" as well.
The line "This... is our Budokan!" The sequence leading up to it had Yui reflecting on how much she'd grown and changed having been in the Light Music Club, rushing to get to the festival with her guitar in hand, then she finally arrives and gets to sing "Fuwa Fuwa Time" like she'd been hoping to several episodes ago. Such a wonderful moment in such a great anime.
Ui: Ganbatte, Onee-chan!
And then there's after they finish the song. Off the cuff, Mugi starts playing her part of its title phrase in a loop. Everyone else looks back. Ritsu joins in, then Mio and Azusa. It takes a while for Yui to absorb, but she joins in too along with the rhythmic claps of the audience, building up to an encore of the vocalized refrain. They were finally having fun with it instead of worrying.
Nodoka: What are you going to do with a cellphone photo?
Azusa: I'm going to send it to Sokabe-senpai.
Nodoka: But do you know her number?
Azusa: That will be your job. *click*
Episode 5, Mugi, Mio, and Ritsu are greatly moved by Yui's guitar performance against a background of fireworks, stars, and ocean.
Episode 7, several beautiful moments between Yui and Ui, but the best is when they receive each other's Christmas presents at the end.
Episode 8, when Ui and Azusa are watching HTT perform "My Love is a Stapler." Ui looks over and notices that Azusa is standing on her toes, absolutely entranced by the band. This is the moment that convinces her to join the club where she meets her 4 best friends.
Episode 13 has an absolutely magical moment when Azunyan finds her soulmate "Azunyan 2", a fluffy little kitten.
By K-On! Standards, Episode 14 ("Live House") had three minor ones.
First, during the preparation-time, when none of the girls in Houkago Tea Time actually knew how to fill out the forms required, Yui quickly befriends another band who helps HTT fill out the forms.
Second, during the soundcheck, everything is going wrong in the beginning, and it's clear that the band loses confidence in their abilities, until the other bands observing them start expressing their support, because they know too well how nervous HTT are, since this is their first public show.
Finally: Ui, Nodoka and Sawako showing up at the venue on New Years Eve, just to see Houkago Tea Time's concert.
Episode 20 of Season 2 has the girls holding their last school performance as seniors. Not only were many episodes building up to it, but Yui has a great surprise when everyone in the audience, her mom and dad, the staff behind the curtain, and her band mates are all wearing the HTT T-shirts. She makes her speech while shedding tears of joy.
The whole of Episode 20 of Season 2 is one of the biggest tear jerkers in the series to date.
The strength and purity of their friendship and love that has grown stronger than any blood-sisterhood over the last three years is perfectly encapsulated in that scene.
And at the very end, when Aki and Yoko have to take a final bow...they do the "Moe-moe Kyun" pose.
In Episode 22, Azusa finally works up the courage to give her sempai her Valentine's chocolate... cake... and they all gather around the window to watch it snow a whole foot outside. The whole scene is the warmest the show's gotten since Episode 20.
There's also the end of that, where Azusa tears up upon seeing a coded text message from her sempai showing that they all got accepted into the same college. It also crosses into funny territory in the scene prior where she prays at a shrine for their success.
The last few episodes are piling on the moments pretty high. Episode 23 has the band cleaning up their room and recording all of their songs on tape to leave for future members to hear.
Episode 24 of season 2 has 2 big ones: Sawa-chan getting thank you messages from her whole class, and the seniors' group song for Azusa (something that was in the final chapter).
The Movie expands upon the song for Azusa. Throughout the events of the movie, the girls are trying to find the time to write the lyrics for it. The song is played again at the film's end, this time focussing on the group practicing for it, showing focus on Ritsu and Mugi trying to sing outside of backup for the first time.
And then there's the second "extra" episode after that. The second half of the episode gives us another two. Sawako looks through her student's yearbook while the others are taking care for her (she's sick at the time.) and then right before the end scene The main foursome walk up on Azusa, Jun, and Ui practicing for next year's opening live session. When they're discovered, the four run away, and we hear the trio playing an instrumental version of Fuwa Fuwa Time as we see the girls leave their high school behind them.
The Death Devil reunion performance, followed by memories of Sawako's high school days, all set to a driving soundtrack of heavy metal in episode 10 of season two. Especially when the bride finally hits the groove.
Season 1 Episode 4: Yui joyously dancing out her idealism under a night lit by smiling stars and laughing fireworks at Mugi's beach mansion, stirring the dream of glory in the heart of the onlooking Mio.
In episode 7 of the second season, Mio decides to thank her fanclub by reading the lyrics of one of the future songs, Tokimeki Sugar. It sounds nonsensical at first, but with a little Fridge Brilliance you can translate what the meaning behind it is — she finds the concept a little scary, but it's okay because it adds to the fun of being in the band and writing songs for them makes the music better.
Mio willingly giving up a free scholarship to a prestigious university in Episode 22 so she can study together with her friends to go to the same university, so Afterschool-Tea Time can stay together forever....and they do, and most likely always will.
Episode 14 of Season 2 has Ritsu trying to help Mugi get hit by Mio. Their plan fails and Mugi gives up on it, but she thanks Ritsu for helping her anyways. Ritsu is flustered by this and hits Mugi herself, yet it still counts as heartwarming because they both end up cherishing it as a happy memory.
In the movie, one of Sawako's strict coworkers (who was her advisor when she was in school) happens to see the girls playing for their class. Instead of shutting it down, he basically admits to their advisor they're quite good and lets them be.