Sonic making his introduction in City Escape, the first stage.
"Talk about low-budget flights! No food or movies; I'm outta here! I like running better!"
Shadow's whole introduction and basically any scene with him, one of few characters to steal the spotlight from the title character.
This game is the first to give us TWO Super Hedgehogs.
This cutscene from Sonic Adventure 2. It perfectly displays the character relationship between Sonic and Shadow and sets a mood that if Shadow isn't stopped here, Eggman will take over the world (Well when played from the Hero's side anyways...) The music just makes it all the more epic.
After organising Shadow and Rouge to raid and detonate an entire island, and with six of the seven Chaos Emeralds in his possession, he declares himself ruler of the Eggman Empire, announces to the entire world that they will surrender or die, and, just to prove he isn't fucking around, powers up the Eclipse Cannon and blows up half of the moon.
The entire "Fake Emerald" scene. Successfully holding Amy Rose at gunpoint aside, he knew all along that Tails had prepared a fake emerald, and pretty much tricked him into admitting that it was the one they were about to hand him in exchange for Amy's safe delivery. And for this little trick, he traps Sonic in an explosive capsule and launches him out of the ARK. Worse still, the capsule's explosion, and Amy's grieving are shown onscreen in the Dark Side story.
Even after the scuffle with Tails that follows this, he still manages to get his hands on the real emerald.
Sonic performs Chaos Control for the first time, with the fake emerald, thus escaping certain death by one of the narrowest margins in a Sonic game.
The opening level of Sonic Adventure 2 is pretty awesome. It's not everyday you see someone freefall out of a helicopter and then board down the side of San Francisco with a part of the wing.
Heck, what about Eggman's first level? The Dark story opens with him charging Prison Island, guns blazing and his theme triumphantly flaring.
Shadow had a very overlooked moment of awesome. He fell to Earth, from SPACE, and SURVIVED. Come on, ultimate lifeform or not, that's gotta count for something.
While everyone (including Eggman) is racing around trying to stop the Space Colony ARK from falling to Earth, Shadow just stares down at Earth, claiming that he doesn't need to help because it's useless anyway. After Amy gives him a speech about how, even though Humans Are Bastards, they can be good too, Shadow finally remembers Maria's true final words: "Give them a chance to be happy!" and quickly runs off to the cannon's core. Meanwhile, Sonic and Knuckles are preparing to do battle with The Biolizard, when suddenly they hear; "Leave this one to me." Shadow walks towards his prototype, calm determination etched into his face, while Sonic and Knuckles stare at him bewildered. Shadow whirls around, declares "I'll take care of this, while you run to get the Chaos Emeralds!", trashes The Biolizard in an epic one-on-one battle, and finishes with "GET OUT, YOU UGLY PROTOTYPE!"
In what could easily be a call-back to the earliest games, Eggman can run at the same speed as Sonic in the Chao World.
Amy reasoning with Shadow to save the Earth.
Amy: I know that people fight over the most trivial things. Some people may be selfish, like the professor said... but they're basically good, if they try their best and never give up on their wishes. They always have a reason to be happy; that's why you should help them out! Saving them is a good thing! Shadow, I beg you, please do it for them! Give them a chance!
Amy's entire involvement in this game is a crowning moment for her developmentally. In Sonic the Hedgehog CD, Amy was captured by Metal Sonic and was a prisoner the entire game. In Sonic Adventure, Amy spent most of her time running away from someone or something until she finally decided to stand on her own two feet and fight her own battles. She then decided to become a stronger person and teach Sonic to respect her and this game perfectly shows just that. She's the first one to jump into action when Sonic is arrested, beating even Tails to Prison Island. With Tails' assistance, she then sneaks into the prison, steals a security card and lets Sonic free. While she does fall into the background a bit until the last story once they get off Prison Island, she's still a fugitive on the run, a contributing member of Sonic's team and, if her appearance in 2P mode is to be taken as truth, her athletic skills have developed from slightly above average with her hammer to full on Badass Normal, with her able to play by Sonic's rules and decently keep pace with him. While she goes overboard in later games and becomes a hotheaded psycho, this game showed perfectly what a little bit of self confidence and determination can do for a girl whose only notable contributions prior to then were getting herself kidnapped.
Tails gets one in the game as well, for showing off that his engineering genius rivals, if not surpasses Eggman's. How does Tails do this? The fake Chaos Emerald of course. Think about it, the Chaos Emeralds are ancient and powerful beyond belief, and Tails MADE ONE. Yes it was fake, yes it was designed to destroy the Eclipse Cannon if put into the core, but in the end it implies that Tails originally made a perfect copy of the emerald and only later modified it. Nobody else is implied to have this much technical prowess.
"Can you hear me, Sonic, Shadow? Everyone here is rooting for you! Good luck, and give him trouble!" - Rouge
When you realise what your reward is for getting all the emblems... well, if you're a long-time fan, that moment will stick with you for quite a while.
Besides being very Heartwarming, having your Chao evolve into a Chaos Chao can be this, due to the hard work of raising it for so long. Also, it's immortal now, too.