Jason stepping up to reassure Tommy after they've freed him from Rita's control. Tommy is aghast at the chaos and destruction he's wrought, but Jason assures him it isn't his fault. "What you did, you did under Rita's influence. You own the Power now. We need you." And as the six of them morph together for the first time, they forge a friendship that will last to their dying days.
It's not so much the Rangers' reaction to The Reveal of Tommy as the White Ranger as much as it is Zordon's reaction to their reaction. While everyone (except Kimberly, who's passed out on the floor) is celebrating and welcoming Tommy back, Zordon smiles(!) and chuckles like a father watching his child's reaction to seeing a new bicycle underneath the tree on Christmas morning.
A very heartwarming moment when Tommy attempts taking the Zeo Crystal, protected against evil (implied to be responsible for Lord Zedd's appearance !) and requiring Incorruptible Pure Pureness : He gets a very serious warning that with his past as the evil Green Ranger, there is no way his heart can be pure enough to claim the Zeo Crystal. He ignores the warning and does it anyway, going through a painful seizure as his past evil deeds flash back in his mind. Even so, he overcomes it all and succeeds in taking the McGuffin for his team.
A subtle one for the baddies in "Fourth Down and Long", where Lord Zedd, Goldar, and Rito are playing a game of football. After all their Villain Decay and repeated failure to take over the world, seeing the three of them just hanging out like normal friends was oddly cute.
In "A Zeo Beginning, Part 2", Zordon officially gives the Rangers their new powers. When he gets to Rocky and says that he is now Zeo Ranger 3... Blue... the camera cuts to Billy (who had just given up his spot as a Ranger) and he sighs... and Alpha 5 puts a hand on his shoulder. Even though it was Billy's idea, that moment showed that the finality of his decision wasn't lost on him. For the viewers, that and the later shot of the old Mighty Morphin uniforms in display cases showed that it really was the end of an era, as Billy was the last of the original five.
If you are a long time viewer, who grew up with the series: in the same two parter seeing the guy that originally said Recruit A Team Of Teenagers With Attitude, Zordon of Eltar, basically allow one of the Rangers he made a mark on to kill him for the good of the universe and unleash the force that would save the day... Tear Jerker and Heartwarming Moments all in one.
In an earlier flashback, the Magna Defender reveals that Scorpius (the current Big Bad) killed the Defender's son Zika. To destroy Scorpius, the Magna Defender causes a huge firepit to explode, threatening an entire space colony in the process. Realising what he's done, the Magna Defender heads towards the firepit to neutralise the energy and save the space colony, and there's a scene where he's crawling through a hellish firepit to reach his goal, his armour damaged and cracked, his cape in tatters - when the scene shifts, and the Magna Defender is calmly walking through a field alongside Zika, who is holding a bundle of flowers. Zika tells his father that he's proud of him, and he gives his father the flowers, and the Magna Defender responds by saying that he was proud of Zika, too. The scene shifts back to the Magna Defender crawling towards the firepit - and he's still holding the flowers that his son just gave him.
And in the following episode Magna Defender's spirit appears to Mike, giving him his sword and declaring him to be his successor, telling Mike it's his destiny.
In the finale, Wes' dad has finally shifted from Screw the Rules, I Have Money! to Screw the Money, I Have Rules!. People are being attacked by the Mooks and he jumps out of his limousine, shoves a mother and her kid inside, and yells at the driver to take them to the hospital. The only time Power Rangers has ever made me cry. Actually, all of Time Force's ending, especially Nadira. She and Trip help a mother give birth in the middle of a store, and her amazement leads her to begin questioning the wisdom of her father's vendetta against humans. She asks Frax, her father's recently-imprisoned Starscream, and he finally breaks down and begs her to stop the cycle of hate. She tries to save a child and is accidentally shot by her father, causing Ransik to make a HeelFace Turn at the realization that he could have lost his only daughter. If he hadn't, he would have won.
The end of "Legacy of Power", the series' 500th episode and a Clip Show framed around a video of the history of the Power Rangers to that point, playing while the current Rangers are trying to find Tommy (who's been kidnapped by the Big Bad.) The scene ends with this:
Tommy (on video): If anything should happen to me, these three (shots of Conner, Ethan and Kira) are charged with protecting the planet.
Kira: That's us!
Tommy: You're in good hands. (video ends)
Conner: He really trusts us.
Hayley: He does.
Ethan: We're gonna have to save him. We owe it to him.