"Rise of the Turtles pt. 2" After recounting the tale of how he lost his wife and daughter, and Leo agreeing that he lost much (he is no longer human), Splinter says that he had gained much as well- his four sons.
April slowly warming up to Donnie in "Monkey Brains" especially coming to his defense when the other Turtles laugh at him for getting beat up.
Also from "Monkey Brains," when Donnie returns to the lair after getting beat up his brothers immediately go to help him and only laugh at him after they are certain he's ok.
After Kirby decides to cover the turtles' escape from the Kraang's base, and, as a result, gets left behind (which is something of a moment in and of itself, since April's watching it all happen). Devastated, April hugs Donnie, who goes on to reiterate that they'll get her father back. It's still on the sad side of heartwarming, but says a lot.
Any scene of Mikey bonding with Leatherhead, starting with Mikey feeding him "pizza noodle soup". Then after Leatherhead talks to Mikey about what a dangerous monster he is, Mikey says "Maybe you just think you're a monster because everyone treats you that way." Maybe he really is wise beyond his years.
Raph's worry when Mikey is knocked out when they're fighting Snakeweed. He doesn't even try to hide his relief when Mikey finally comes to.
Also, Raph shouting Mikey's Catch Phrase "Booyakasha!" as he delivers the final blow to Snakeweed, for the same reason. Doubles as a Moment Of Awesome.
Especially heartwarming when, back in the layer, Raph says that he has no problem risking his own life, but not his brothers.
Karai choosing to help Leonardo escape from Snakeweed instead of letting him die. Especially notable since Shredder had already given her a stern warning that she wasn't to let him live the next time they met.
Leo saving the knife she threw at his head as a memento at the end of the episode.
Mikey and Don's concern for Raph when he gets bitten by Xever/Fishface.
Especially the way Mikey defends Raph, busting out his most badass moves, often not even taking his eyes off of Raph while fighting off all comers. Doubles as a Moment Of Awesome.
Splinter decides to give April her own ninja weapon, a Japanese war fan. Also heavily overlaps as a tearjerker since Splinter originally planned on giving it to Miwa.
Karai saving the Turtles, and Leo saving her in return.
Despite the rocky moments and strained relationship we've seen between Karai and her father, she still immediately moved to save him from the Turtles' attack.
Donnie's concern for April is palpable while she's being chased by Karai.
The turtles are clearly impressed and happy hearing about April's escape from Karai.
Splinter afterwards says one of the most important lessons about being a Ninja is making it back home.
When April is fighting Karai, she says she lost her mother. Karai pauses the fight and is caught off guard, and when April is escaping, Karai gives her a...genuinely sorry look.