One absolutely beautiful moment in "Music" is when Didi rocks Dil to sleep, singing "Beautiful Dreamer" all the while. This is after an entire episode of the babies trying and failing to get Dil into music and imagine stories along with it. As Didi continues singing, Dil starts dreaming of swirling lights and colors. In the last seconds of the episode, the shape resolves into an image of Tommy's face.
Tommy decided not to abandon Dil with the monkeys in the movie.
"Moving Away", the one where Angelica and her parents are moving. Angelica and the babies are all happy about it... except for Tommy, who says it's terrible and goes on to tell the story of how Angelica brought them together and it's because of her that they're friends. Angelica hears the whole story and is in tears afterwards and doesn't want to move. Probably heartwarming (IMO) because it shows Angelica's "heart of gold" side.
Chas, Kira, Chuckie, and Kimi being a happy family.
Specifically, the first episode after the second movie, where Kira formally adopts Chuckie and Chas does the same with Kimi.
For this troper it's this one line in the second movie from Kira:
"Chuckie, how would you like to dance with your new mommy?"
Kira's speech to Chuckie in the episode "Changes For Chuckie"
"Hi Chuckie. Well I guess washing Wawa wasn't the best idea. You know there is still a lot about you I don't know yet. Like your favorite sandwiches or your favorite games or the songs or stories you like. Those things may take a while. What i do know is I love you very much. I hope that will do till I get the other things sorted out."
The song sung between Tommy and Angelica during the All Growed Up TV special, as it shows clips of all the adventures they had as infants. Considering this was going to be the grande finale of the series, it was pretty poignant.
Angelica saving the babies from Josh, also doubles as a Moment Of Awesome. "They may be just a buncha dumb babies, but they're my DUMB BABIES!"
The Rugrats second Christmas special "Babies In Toyland": The babies find themselves in a Nativity scene and think the baby Jesus is real. They feel sorry for him since they believe Santa isn't coming and he's lying alone in a pile of hay with no gifts, so one by one, they each give him their own gift just because they want his Christmas to be special. When they found him, the figure's expression was neutral. When they're done, it's smiling. Awwww to the power of 10,000.
"I'm a big brave dog." Whenever Chuckie comes out (inevitably nervously, and to himself) with this carefully-deployed catchphrase/mantra, a Heartwarming Moment generally won't be far away.
The episode where the kids go to see the Dummi Bears on stage, but Angelica can't go due to a cold. The babies return, seeming to have caught Angelica's cold, but they were only faking because they wanted to be with her. Especially sweet when you remember how much Angelica bullies them. And everyone still got to see the show because Stu bought the video of it while they were there.
Phil and Lil reuniting in "Together At Last".
The episode "Lady Luck." Grandpa goes to bingo night at a senior home, and due to his ramblings, the babies think he's going to meet a friend called "Lady Luck." They search all over the home looking for Lady Luck, in the process, turning the soporific home upside-down and helping all the elderly visitors to have a lot of fun. In the end, the previously-tough operator of the center, seeing how much fun the seniors are having, both becomes nicer and goes out with Grandpa for ice cream.
The end of the Mother's Day episode with the poem Chuckie's mom wrote for him.
My Sweet Little Chuckie
Though I must leave you behind me,
this poem will tell you where you always can find me.
When a gentle wind blows, that's my hand on your face.
And when the tree gives you shade, that's my sheltering embrace.
When the sun gives you freckles, that's me tickling my boy.
When the rain wets your hair, those are my tears of joy.
When the long grass enfolds you, that's me holding you tight.
When the whippoorwill sings, that's me whispering, "Night night."
The episode where Susie's tricycle goes missing. She accuses Angelica of stealing it because Angelica had been mean to her earlier (which makes the assumption somewhat rational). As the episode goes on, it looks like Angelica did steal it and Susie ties her Cynthia doll to a balloon for Chuckie to let go as revenge...only for it to be revealed later that Angelica didn't steal it after all. Susie's father had moved the trike so he could pain the shed. After this, Angelica reveals she was making (well, trying to anyway) an apology card for Susie for being mean to her earlier. So we have an episode where Angelica, of all people, is apologetic. And to top it off, she does get her doll back.
Spike nursing kitties in the episode "Spike's Babies" and the babies try to help him.
The ending of "Spike Runs Away." A man returns Spike, who's been missing for what appears to be weeks, late at night. Stu says they need to wake Tommy up, but Didi says it can wait til tomorrow. However, as soon as Spike hears Tommy's name, he rushes to his room and jumps on him in the crib. Tommy happily squeals "SPIKE!" as his dog licks him over and over again. Sweetest reunion between a boy and his dogever!
In "A Step at a Time", when the babies are recounting the time they first walked, Tommy tells them that when Chuckie had come over to his house with his dad (this was after Chuckie started walking), and Tommy wanted to play with Chuckie, so he pulled himself up using the back of the couch in the living room and starts walking. Chuckie is the whole reason why Tommy started walking in the first place.
In "Angelica's Worst Nightmare", Angelica wakes up from a nightmare involving her newborn baby brother terrorizing her and her parents ignoring her. When her father goes to see her, she starts breaking down and telling her dad that she doesn't want a new sibling and to be ignored like she was in her nightmare. Her father reassures her that he and Angelica's mother will always love and care for Angelica since she is their first child and she will never ever be replaced.