Watching all the Red Rangers from Jason to Cole morph in the episode "Forever Red" never fails to send a chill up and down the spine.
Seeing Jason casually walk up to the leader of the villains and flatten the guy with just his bare hands (while every other Red Ranger present except for Tommy was using their personal weapons), all the while teaching the current Red Ranger a thing or two about being a Ranger... there are no words.
Jason's line "Ya know, if you miss King Mondo that much, I promise we can help you join him" is another great, casually badass moment for him.
Tommy Oliver's kicking ability. First, during the brawl with the Cogs, Tommy thrust-kicks one hard enough to send it flying through the air several feet and into some machinery, with enough force to send sparks flying from it. Remember, the Cogs are robots. Made of metal. And Tommy was UNMORPHED at the time. Then, after morphing, he engages one of the machine generals (alongside Wes) and finishes it off by delivering a flying kick so powerful that IT EXPLODES.
How about the very end of the episode, that is also a Crowning Moment of Funny? After Tommy leaves, and Cole calls him the greatest Ranger, the others immediately start playing a game of one-upmanship as they try to put themselves over as the best Red Ranger, all taking shots at Tommy in the process. Even Carter gets one in, mentioning his hair not being "up to regulation."
The more recent Rangers are asking what possibly could the Bad Guys find on the Moon, and Jason says "Serpentera", looking at Tommy, who nods. You can tell that if it were just Tommy and Jason, they wouldn't have to explain how much trouble they are in.
How about even earlier in the episode, when Jason enters? Doubles as a Crowning Music of Awesome.
Just the way they portray the strong brotherly relationship is this. A bit of casual banter followed by that handshake...
Tommy: I was starting to wonder whether you would show up at all.
Jason: You didn't think I was going to let you do this without the original Red Ranger, did ya?
Danny got his Moment of Awesome in the episode "Three's A Crowd". Danny's in love with Kendall, the girl who works with him in the flower shop and who is also being pursued by a Jerk AssRomantic False Lead; when they are all attacked by the Mooks, Danny, knowing that he can't let them see him morph, seems to sacrifice himself to give them time to escape. He thinks they've left (they went for another door) and as the girl and the Jerk Ass watch (this is a beautiful Oh Crap! moment for him), he morphs in front of them and starts clearing the room with the bad guys. Weapons-grade made of win moment; also, if you look at the girl and pay attention to Danny's last name... you get the distinct impression that you're looking at the parents of 'Z' Delgado from Power Rangers S.P.D.
Approximately 105% of Zen-Aku's scenes.
And they found a pretty badass way to beat him getting the rhino and armadillo zords and using them to super soccer kick the Predazord when it was about to fire it's predator wave.
Combined with a crowning moment of heroism when he nurses Alyssa back to health after she was seriously injured, then stops a bunch of other villains from hurting her. It's no wonder that he got resurrected at the end.
The fight between Cole and Super Powered Toxica and Jindrax. The whole fight starts after Nayzor is destroyed and consists of the villains suddenly turning very serious, and throwing around huge beams and explosions. Some of the fighting moves were also very impressive. It then goes right into the two trying to drop Cole to his death, but he uses his Battlizer to survive and then turn the Curb-Stomp Battle to the other side, with the Orgs receiving it.
There is that one scene from "Soul of Humanity" when the construction workers came back to the collapsed building to save two girls and a dog, which was right after Mandilok was calling all humans selfish, basically putting a foot in that giant mouth of his.
The Rangers indirectly going up against Animus in "The Soul of Humanity" would certainly qualify.
Let us emphasize this moment: The Rangers have no powers, their Zords have been absorbed into Animus, the being who created them and a god, and these puny humans whom Animus felt did not deserve or appreciate life are all calling their animal spirits and basically saying "Hey! What do you know?" and "YOU SUCK!!!" And as an audience, you realize that these guys have nothing that can stand a chance against this god of living things. To top it all of, Merrick decides to stay on Earth to help his friends rather than joining Animus.
The team-up episode with Time Force. The previous two team-up episodes had received some flak from the fandom, the Lost Galaxy/Lightspeed Rescue team-up having had several behind the scenes problems and having used the Sentai team-up instead of being a budget-buster American produced episode, and the Lightspeed Rescue/Time Force team-up having been shoved into a single half hour and involving more than the usual amount of Fridge Logic. But "Reinforcements From The Future" showed they'd figured out this team-up thing, managing to not just be the token team-up but actually wrap up some dangling loose ends from the previous season and spare some time for team-to-team interaction.
One good mention: Jen and Alyssa using the Chrono Blasters to fight off a few Putrids.
Cole's showdown with Dr. Adler/Master Org, ending with him absorbing Master Org's power and sending it right back at him. He then tops it off by refusing to sink to Adler's level and kill the now defenseless man.
"A Father's Footsteps. Alyssa catching Samurai Org's sword strike in a Bare-Handed Blade Block and then proceeding to break the sword with her elbow. Her father had every right to be impressed.
When all the Wild Zords destroyed by Master Org along with all the Wild Zords that had been lost for centuries returned in the Ranger's Darkest Hour, giving them more power than ever before, allowing them to destroy the now-literal monster Master Org once and for all and end the terror of Orgs forever.
Before that there is the Rangers, powerless against Master Org, saying that while they have lost their powers and their Zords, they are not giving up until they die.