In "Final Curtain", Spidey takes on the Green Goblin, his mini-army of "Pumpkinheads" (armed to the brim with high-tech weapons, in fact), while going through a booby-trapped city street,all by himself. Whoa...
Let's be specific here, Spider-Man is attack by a platoon of enemies arm with Hand Cannon, an entire neighborhood of automated Turrets that fire GRENADES...with the firing rate of a machine gun!(yes you heard that right), and the Goblin's own mini-air force and they didn't even slow him down, he defeated them all. Heck, the only thing that even TOUCHED Spider-Man for that matter was the Green Goblin himself, even then all he really did was mess up his costume.
Wait, we mentioned "Final Curtain" before "Gangland"? That entire episode was pretty much a battle royale between Doc Ock, Silvermane, Tombstone and Spidey. Tops it all off with Opera (specifically Rigoletto) playing all throughout the entire fight and Spidey finishing the fight with Tombstone down in a sewer.
Green Goblin backflipping over a building he's about to crash into, and landing square on his glider as it comes out the other side. Best of all is Spider-Man's reaction: "Okay, wow. Just... wow."
Both times Peter fights off the symbiote's influence, first with the help of the memory of his Uncle Ben, then by summoning forth all the positive emotions associated with the people he loves. It goes to show that Peter's real power doesn't come from his spider-strength or his web shooters, but from his heart.
In "Group Therapy," Eddie takes Mary-Jane on a high speed motorcycle ride just because he thinks taking her out will hurt Peter. MJ doesn't take any of that crap.
Mary-Jane: Pete may not be perfect, but he's twice the man you'll ever be.
Also in "Group Therapy," Aunt May standing between Spidey and the entire Sinister Six. That woman has guts. Also a heartwarming moment both in story (defending her nephew without even knowing it) and from a meta perspective.
Also in "Group Therapy," the Symbiote defeating the Sinister Six all by himself is quite awesome ; at first the Six wipe the floor with him, crack some jokes, the mask's eyes widen in a "Oh Crap" moment... And then he proceeds to defeat each and everyone of them, showing greater strength and agility than Peter himself.
In "Nature Vs. Nurture," the entire football team works together with Spider-Man to save Gwen from Venom.
Spider-Man: Thanks for the assist. Flash: Anytime. Now get that creep!
The (temporary) team-up of Spidey and Rhino in "Accomplices".
Vulture is holding Norman Osborn high above the city and, despite the peril he's in, continues to berate his captor.
Osborn: Toomes? Vulture: Not Toomes, now! I'm what you called me, I'm the Vulture! Osborn: I called you a buzzard. Vulture: What? Osborn: You can't even get the name right.
Tombstone utterly schooling Spidey during their first encounter in "The Invisible Hand".
This far and no:
Electro: I'm ELECTRO!
From the same episode, Peter telling off the snotty cheerleader he was trying to tutor. For anyone who's ever been blown off by the popular crowd before, it was quite satisfying.
Peter: Look, this tutoring thing is something you need, not me, so when you're ready to learn, lemme know. (goes off to fight crime)
Poor, meek little Otto Octavius has just survived a lab explosion. His abusive employer, Norman Osborn, arrives to pull him out of the rubble and starts to berate him yet again, when Otto's eyes narrow and...
Doctor Octopus: "Silence you imperious moron!!" (SLAM!!)
Comic homage: In Amazing Spider-Man 33, there is an iconic image of Spider-Man using every last ounce of strength to lift a bridge off of himself. It's been repurposed for the show, with a huge light fixture in place of the bridge, and it couldn't be any more awesome.
Peter: Can't... budge... it... so cold... muscles stiff... no... This is it. I fought the good fight... did my best... foiled Doc's plan. No one can ask any more of me... (sees Gwen lying unconscious nearby) What am I doing? I can't give up, not with Gwen depending on me! (lifts the pile of debris off of him)
The entire episode of "First Steps" is a Crowning Moment of Awesome for the Sandman, who not only demonstrates what a man who is made out of and can control sand can really do, but even manages to muster a few Pet the Dog moments, topped off with a Heroic Sacrifice... which he survives. He built up too much coolness to be killed, and the last look on his face in the episode shows it.
The epic scene where he leaves Spiderman wondering "How am I supposed to beat up a beach?" The ensuing Colossus Climb is a thing of beauty.
"Accomplices" turns out to be a whole-episode CMOA for Norman Osborne, who caused hell for his enemies, makes five hundred million, and still has the Rhino data, all with literally no risk to himself.
"Probable Cause". Spidey derails a train with nothing more than a well-placed cart full of gold bars. Also, he fought like hell against Shocker and the Enforcers.
For Harry Osborn... "Harry, what are you doing?" "Cowboying up!" Even his ass of a father was impressed. The Chameleon who was disguised as his father at the time, however, seemed a bit frightened.
What about Norman's reveal, he managed to be cool and scary... In his own sickening way of course.
And another one I forgot, to me, this Goblin one-liner was awesome.
Green Goblin: We all wear masks, Spider-Man, but which one is real? The one that hides your face? Or the one that is your face?
Which becomes even more awesome in hindsight, after the Goblin's real identity is revealed.
A rather off-screen example, Gwen got her father into going to Thanksgiving with Peter. Here are the details:
Captain Stacy: She insisted on coming. Gave me the look.
Doctor Octopus spends much of his introductory episode insulting Spider-Man's intelligence and gloating about his own genius... which makes it incredibly awesome when Spidey suddenly out-gambits him with an almost pathetically simple trick.
In the episode Natural Selection an old woman gets a real great Crowning Moment of Awesome. Spidey is fighting the Lizard in a crowded train. Lizard has Spidey on the ropes, yet that is stopped by the woman smacking him with her purse, allowing Spidey to gain the upper hand.
You know there is something wrong with you when Flash Thompson of all people gives you a What the Hell, Hero? speech.
Spidey's first fight with Kraven turns into a complete curb stomp in Spidey's favour, with the webslinger explaining Kraven's difficulty in this hunt being because he's a Spider-"Man" and not a beast.
Spider-Man: You just. Don't. Get it. You're good. Maybe at the whole hunting thing you're even the best. But, when it comes to the New York superpowered scene *catches Kraven's knife out of the air with ease* you're out on your league.
Spidey's SECOND fight with Kraven is just as awesome. Despite the fact that Kraven actually managed to keep up since he spliced animal DNA into himself, the second half of the battle has him curbstomped when Spidey makes full use of his web slingers, trapping Kraven inside a web cage and then compensating for Kraven's newfound speed with repeated spamming of his webs, ending with an epic No-Holds-Barred Beatdown and a Badass Boast.
'Spider-Man: Take that pussycat! A lesson in humility courtesy of the original and STILL number one genetic misfit: Me.'
The entirety of "Gangland", but especially when Spidey actually manages to take down Tombstone, and reveal his true colors to the public.
Both of the Venom arcs are chock-full of awesome moments. From the first arc, the symbiote hijacking Spidey's body and giving the Sinister Six a whoopin', Spidey's mental fight against the symbiote, Venom beating down on Spidey in his own back yard, Spidey fighting off Venom right outside Aunt May's hospital room, and their fight at and around the Thanksgiving parade. From the second arc, Eddie's Paranoia Gambit, and their fight in the school.
Mary Jane gets one in her own way. When Peter has to leave her at the dance, Sally takes the opportunity to try to mock her for it. Mary Jane responds by effortlessly getting Randy (Sally's boyfriend) to dance with her instead.