In episode one, Genki kicks a dinosaur in the face to protect Mocchi. The fight brings down the whole building. And that is just episode one.
In episode 4, Genki and Suezo work together to squash one of the Seed Sisters with a rock, proving early on that both of them could be capable in battle.
Tiger's rematch with Captain Dino in "Tiger of the Wind". Despite being injured from their last encounter, Tiger takes him down with one strike.
Genki punching Pixie in the face. Even in the dub, where the actual hit was edited out, we still see it in first-person view and Pixie clutching her face in shock afterwards.
Pixie's response to that was pretty awesome, too. She zapped them off a cliff.
And then Genki and Mocchi climbed back up.
In Episodes 10 and 11, the gang organizes a mass breakout from Pixie's slave camp. Even when the majority of the Searchers were captured, they just broke out from the inside.
The ensuing battle with Pixie had an awesome moment of teamwork, too. Golem stepped up to hold off Big Blue, while the others regrouped to attack Pixie all at once. She zapped Suezo and Hare, and dodged Tiger's Blizzard...but couldn't counter everyone and was left open to Mocchi's Cherry Blossom Blizzard.
Even cooler was that it showed them learning mid-battle. Tiger tried to attack Pixie and Big Blue solo and was knocked away. Then the searchers attacked one after the other, which still didn't work; what Pixie couldn't handle Big Blue could. Seeing his friends in trouble, Golem attacked—unexpected, as he was a pacifist, and he used a never before seen attack—to stop Pixie's chariot and free the rest to attack her by herself. Then they all attacked at once.
The gang rescuing Holly from Moo. Holly herself deserves credit as she refused to give in to the monster.
The gang's battle against Guji, a Godzilla-sized Zilla monster. They defeated him by tying him to Captain Horn's ship and punching him into a whirlpool.
Pixie got quite a few, being one of the few monsters who could damage Moo. But her biggest had to be in the episode "My Name Is Pixie". She rescues Genki from Scale Jells, but is hurt saving him. He saves her from quicksand as they try to return to the group, all as Pixie learns to overcome her differences with Genki. However, they're cornered with no chance of escape, as the Scale Jells have a Healing Factor. The Captain orders her to destroy Genki to save herself, but she refuses. As the Jells power up beam cannons to kill them, Pixie prepares an attack.
Pixie: I take orders from no one! I only fight for what I believe in! My name is...My name is...My name is PIXIE!*completely obliterates the Jells*
Mocchi taking out Gali with one shot of the Mocchi Cannon.
Tiger running as fast as he can to save Holly from a Black Worm's poison. Even after he was poisoned himself.
Genki gets one in that same episode when he opts to fight some Black Worms chasing them to allow Tiger to go on ahead. When Tiger comes back, he finds all the Worms beaten (not to the point of turning to Lost Disks but still pretty wounded) and Genki bruised but still kicking.
Holly got one too. The reason she got poisoned in the first place was because she shoved Tiger out of the way of the dart.
The Centaur Warriors of the Colosseum vowed to protect it from Moo's baddies, even though no one was left to visit and the baddies would likely return even stronger. Genki was so impressed by their devotion he cried.
Captain Jim was a Person of Awesome. He fought Arrowheads with harpoons and the thrusters of his boat. While shirtless.
In the finale, he's seen leading an army of rebels against Moo's forces.
In "The Town That Disappeared", Bajarl has captured everyone but Holly and Mocchi with his magic jar. This leads to a fourfold moment of awesome.
The titular town's mayor had rigged explosives as a trap for Bajarl. Even when it was unable to be used before everyone was gone, he hid them away along with a diary and map, with his last wish being someone find it and use it.
Mocchi decides to distract him so Holly can get away, getting hurt in the process but still willing to fight.
Holly goes and uncovers the explosives and lights the fuse, having to both hide from Bajarl and get back quickly, and lift up several mounds of rocks to find the explosives.
Tiger, while inside the jar, uses Blizzard to freeze Bajarl, acting as a further distraction and keeping him from escaping as the town blows up.
This is made even more awesome by the You Shall Not Pass moments Pixie, Big Blue, Golem, Hare, and Tiger get to make sure Genki, Holly, Suezo, and Mocchi reach Naga.
Then Holly, who's generally a Non Action Girl, pulls a dagger on the monster, even in the dub. And this whole Storming the Castle thing? It was Suezo's idea. And his original plan was to attack the castle by himself. To top it off, his Teleport skill works both to get to the castle and in combat, and he actually damages Naga during the fight.
Suezo: No. Let me get 'im. He got my sister...and the rest of the villagers as well!
Suezo got another awesome moment in the uncut Japanese version, where to protect Holly, he bit down on Naga's arm so hard it was rendered unusable. It was likely cut due to blood and that he had to be peeled off of him.
Genki and Mocchi use the Mocchi Cannon to blast Naga from behind, hitting his injured shoulder and launching him over the edge of the parapet.
Holly knifed Implacable Man End Bringer, short circuiting the robot and giving her time to get Genki to safety.
Tiger tricking Lilim so he and Jagged Hound could end their duel safely, and protect the baby Jagged Hound was guarding. It ends with well-deserved revenge on Lilim.
Lilim got a villainous one in "Magic Stone Mayhem" where she outsmarted the heroes at their own game. Even though she didn't kill them all-and that was only due to Captain Weed's interference-she did capture the Magic Stone.
In "The End of Duarahan" it's implied Suezo teleports the team to their final goal, one ten meter hop at a time, over a day, without taking a break.
The Weeds got one in the same episode when right as the heroes are about to recover the Magic Stone, they snatch it up to deliver it to Moo.
Durahan ensuring Moo couldn't kill him even though the latter clearly outmatched him, as he had the Magic Stone.
Durahan taking revenge on Lilim right before his ship explodes.
Moo taking whatever the heroes can throw at him, and being defeated or retreating while averting Villain Decay. Helps that he was made to be a Super Soldier.
The second season finale. Yes, it was very sad. But it also had the Searchers' quest finally be resolved, them finding the Phoenix, and Mocchi, Suezo, Golem, Tiger and Hare willingly sacrificing themselves to become the Phoenix and bring back a peaceful world. That takes pure courage.
In the uncut episode 48, the Phoenix takes Moo's strongest attack—basically a nuke, which leveled an entire mountaintop— without flinching. It then latches onto him and starts scorching him.
In episode 69, Mum Mew defeats an entire gang of robbers by herself, proving she can be awesome even if she performed a good act when she meant to be evil.
The entire show is awesome for the sheer amount of effort put into the story, animation, and voice acting.
The fact that across the three seasons, every species of monster from both games was seen aside from the Doodle and living Disk.