In the episode "Memoirs of a Finster" at the end when Kimi was doing her family tree.
Though it's kinda sad in its overall context, Chuckie's comment of "Why are you on the phone with Dad when he's right downstairs?" when Kimi is on the phone with her biological father is sweet considering it showcases that he doesn't use the label "step" to describe his relationship with Kira and Kimi and that normally Kimi doesn't use it for hers with him or Chaz.
In the episode "The Finster Who Stole Christmas" when Kira and Kimi got back from Japan and arrived home to Chaz and Chuckie.
They all were so happy to see each other. A lot of great family hugs these four shared.
The end of "Curse of Reptar," where Tommy dreams about playing with Reptar Jr. as a baby and then goes outside to retrieve his old toy from the construction site.
This would especially count as a moment of heartwarming to fans of the show whom saw the original, which would be the majority of watchers.
And the reason Chuckie remembers Reptar more fondly than the others? It's because of the doll that he gained a new mother and sister.
The end of "Trading Places."
Chaz and Kira celebrating their 10 year anniversary in "Petition This!".
Anything related to Chuckie and Kimi's relationship. Episodes such as "Bad Kimi" and especially "Memoirs of a Finster" show just how much these two really care about each other.
"Bad Kimi" seems to have one for Kimi's friend Z. Z throughout the episode was seen as a Jerk Ass due to Chuckie's assumptions, to the point where he thinks Z has convinced Kimi to steal. When it's revealed that wasn't the case (The stuff was actually being taken to a shelter), Chuckie pleads for Kimi to forgive him. Kimi instead snubs him, and Z actually is shocked, finding it disgusting someone as nice as her coldly dismisses Chuckie like that due to his justified uneasiness with him.
"Susie Sings the Blues": Susie has been told by a "music producer" that she has to pay $1000 if she wants a studio session. Unable to come up with the money on her own or ask her parents for it, Susie turns to her older sister, Alisa, who has been saving up for a house. Alisa agrees to give her the money, however, Susie soon discovers she was conned. Completely devastated, Susie goes home and tells her sister about the lost money - and Alisa is more concerned that her sister put her life at risk than she is about the lost money.
Then Alisa suggests that Susie forget about singing and then Susie gives a speech about how singing is who she is, before remembering the talent night at the Java Lava. She arrives and puts on a great performance that gets her a standing ovation from everyone there.
"Sueise's Choice", where Susie has to decide between a prestigious talent show or a language competition, during which she discovers her own mother was a successful singer too. She ultimately ends up giving up the talent show but still performs at Betty and Howard's reception. She then calls her mother up on stage and she and Lucy sing a duet...which quickly turns into a CMOF when we discover that the supposed Noodle Incident between Phil and Lil was that Phil had spray-painted his skateboard over Betty's wedding dress.
The ending of "Lucky 13" - at Angelica's birthday party, Savannah asks if she can sit at the 'Cool Kids' table but Angelica says all the seats are already reserved - for Tommy and his friends. While at the table, Tommy tells her about their plan to save her party and Angelica wonders why they'd do that for her when she'd been so mean to them in the past. Tommy tells her that she's still family to them, and Angelica is very touched.
"Miss Nose it All" has Angelica being forced to volunteer in a hospital but she ends up rushing through her duties because she wants to go shopping, including not reading a story to a little boy. She ends up with a broken nose and pretends to be even worse so she can stay in the hospital and not be seen. While there she learns that the little boy has been sick for a long time and can't go home because of it. Realizing how foolish she'd been, she offers to read him the story again.
In the same episode, when Harold hears Angelica has a very serious disease, he pretty much breaks down in front of her; Angelica, for once, realizes how much Harold really cares about her, and actually feels guilty for making him feel this bad, so confesses him it's just a lie, much to his relief.
Pretty much every time Angelica and Susie show how much their friendship means to each other.
Most notably in 'Susie Goes Bad Lite', when Angelica risks her life to save Susie's, and they both have to be rescued by firefighters; even more, Angelica, who, for once, did nothing wrong and was just trying to help Susie who, for once, was the one doing something dangerous for popularity's sake, ends up Taking the Bullet, saying the whole problem was her fault, so Susie's permanent record can remain spotless.
Before that, when Lil and Kimi find out Susie is going to risk her life, their first instinct is getting Angelica's help instead of their brothers or parents, showing that, despite the problems between them, the girls know Angelica is trustworthy when it comes to serious issues.
In What's Love Got To Do With It?, Susie warns Angelica against keeping her crush on their substitute teacher alive and consoles her when it turns out said teacher got a permanent job elsewhere. Angelica even asks her to stay with her for a while as she bawls her eyes out.
The end of 'In The Family's Way'; after spending a whole week living with Susie's family, Angelica convinces her parents to spend more quality time themselves, starting with cooking dinner together and later chat while eating, something they hadn't done in a while. Charlotte and Drew, once they recover from the surprise, really like the idea.
The end of "Saving Cynthia". Angelica and Susie spend the whole episode trying to get Angelica's old Cynthia doll back. Angelica's rationalization is that Cynthia was a friend that was always there for her, despite being a doll. They have Cynthia back at the end and Angelica nervously starts to say something about "a friendship that was always there". Susie turns around and says "ours?". She says this one word with a smile and a tone of voice that tells Angelica that yes of course they are friends and always will be.
As hard as it is to believe, seeing Cynthia again is pretty darn heartwarming if your a veteran fan of the original series. Angelica's feelings of nostalgia could even mirror those of the viewer.
Tommy and Rachel's scenes together, they were a cute couple while they lasted.
He also got some good moments with Kimi in "TP & KF".
Lil spends most of the second episode being annoyed at Phil embarrassing her. When she goes to a party with friends, the friends talk about how horrible Phil is. Lil then defends him and angrily calls them out for their comments - walking out of the party and immediately going home to apologize to Phil.
In "River Rats", Chuckie tells Tommy that having Tommy around makes him feel like everything is all right, making Tommy afraid to reveal his aquaphobia. When Chuckie finds out about said aquaphobia, his anger about the secret quickly evaporates when Tommy explains that his fear makes him feel like he's letting Chuckie down. Chuckie reassures Tommy he doesn't think less of him and is responsible for Tommy being able to gather his courage enough to save Phil when Phil falls into the river.
In "Dude, Where's My Horse", Angelica bonding with a baby ostrich that accidentally imprinted on her, to the point of going Mama Bear for him several times and at one point giving him a little pink ribbon to wear. Becomes a slight Tear Jerker when she ends up having to leave him behind.
The ending of "Truth or Consequences" is filled with them. Tommy successfully mends his friendship with the others by splicing footage from old home movies of the gang into his recordings of their embarrassing secrets, which even causes everyone to end up on better terms with each other. We also have Didi and Stu state that they are proud of their son Dil even if he is very eccentric.