An early episode has a bit of Fridge Heartwarming. The Rangers summon the Megazord in order to save a little girl from being dropped to her doom by the Monster of the Week. Said monster is still human sized, mind you. One of Zordon's rules is that you can't escalate a fight unless Rita forces you to. Saving a little girl is equally important to the Rangers as stopping a monster from destroying the city.
Rito: Well, in keeping with the holiday spirit, I want you to have this. (hands Goldar a present)
Goldar: Uh, you mean this is for me?
Rito: Goldar, happy holidays, ya big lug!
In "The Wedding," Rita uses a Love Potion on Zedd. Goldar can't believe it and eventually, in "The Potion Notion" learns about it. He hits Zedd with a counter-potion... that has absolutely no effect on Zedd. Turns out, old Ed really loves her. It probably helps that Zedd had realized by that time that he was having as much trouble getting rid of the Rangers as Rita had in season 1, and thus really had no justification in banishing her back to the dumpster in the first place.
Also, Rita originally gave Zedd the potion to eventually overthrow him. But somewhere along the line, up into that point, she comes to truly love him.
Tommy's letter to the team (and Jason, particularly) in "Missing Green". He misses the other Rangers. He tells Jason it's not his fault for losing Tommy's power and doesn't blame him for it; he never had. Jason was able to finally get over his guilt, and when he wins the tournament, he holds the trophy up and shouts "This is for you, Tommy!"
In "Ninja Quest," Bulk and Skull graduate from the Junior Police Academy. Seeing the Power Rangers and Ernie attend the ceremony to applaud them for all their hard work definitely gave this troper some warm fuzzy feelings.
Doubles as a Crowning Moment of Awesome. After disposing of Nimrod in "White Light Part II." The Red Dragon Thunderzord walks across the field and grasps the hand of the Mega Tigerzord and gives him a thumbs up and they nod before striking a pose. While they often do this in later episodes as well, on this one its as if the Rangers are telling Tommy 'welcome back' and Tommy saying 'it's good to be back'.
A Meta one - at C2E2 2013, Jason David Frank appeared for a panel & fan Q&A. Amongst the questions asked where if he would return for the 20th Anniversary season & his assertation that he'd do it for $1, but nothing could top his answer to the first question: if he had ever felt ashamed over his ties to the show.
"Visiting kids in the hospital who has a dying wish to see you? No way, man. To experience that Ė forget cameras, fame, forget all that. Iíve never been ashamed of it. For 20 years Iíve been promoting it and Iím still doing it. Itís part of who I am.Ē
In "Potion Notion", the monster of the day has been putting people under love spells using the same potion Rita used on Zedd, Kimberly gets hit with it and falls in love with Skull asking Skull to the dance and swooning over him, Skull of course has had a crush on Kimberly since day 1. Of course an antidote is released before the dance and Kimberly ditches Skull. The crowning moment of heartwarming comes at the end of the episode where at the dance Kimberly sees how bad she hurt Skull who is sitting at a table by himself, and goes up to ask him to dance. And considering how much the show treated Skull like a buffoon and a joke character it was nice to see them treat him as an actual person with feelings.
In fact, that was the moment that B&S stopped being down right antagonistic towards the Rangers. They still bantered back and forth, but from Potion Notion on you could sense it was more friendly banter than hateful.
It's even more Heartwarming if you believe the theory that Kimberly is Spike's mother.
In "The Ninja Encounter, Part 1", Bulk and Skull try to save the runaway baby, only for Kimberly and Aisha to beat them to it. Though they do get to babysit when Jacob's dad was kidnapped by Goldar.