The opening alone is just awesome and really gets you excited before you start playing. Open Your Heart really adds an extra layer of epicness to it, as well.
Sonic's introductory scene in his story, wall jumping from one tall building to another as a blue blur with his theme playing until he gets to the top of one, after which we finally get a clear look at him in 3D for the first time.
(English): Aww, yeah! This is happenin'. (Japanese): (whistles) Greeeeaaat! I'm feelin' good!
When Tails launches the Tornado I by making the palm trees bend over in what can only be a direct homage to Thunderbirds. Sonic's expression says it all.
Both Sky Chase levels are both pretty damn epic, getting to blow up dozens of fighter jets and dodging bombs from the Egg Carrier is quite fun. The second level is particularly cool, especially when halfway through Tails transforms the Tornado 2 and you fight an insane amount of enemies culminating in a boss fight against the Egg Carrier itself.
Speaking of Tails in Sonic Adventure, who can forget him psyching himself up to fight Eggman alone... then doing it, saving Station Square all by himself? This seems to be the rare piece of Character Development in the franchise to stick, too, as he does so a couple of times in the next game as well.
Eggman: So you beat me to the missile, you little pest...I will make you all pay for this! (Eggman flies up and drops onto the Casino District with the absolutely gigantic Egg Walker) Eggman: You fool! Away, before I make mincemeat out of you! Tails:I'm not scared... I'm not scared! I can do this!
The music in this game. Sonic Adventure has personal theme songs per character and level, showing the different styles of each song. This soundtrack is seen as one of the greatest soundtracks of all video games.
There was Amy's story. After spending most of the story getting kidnapped and running away from a Killer Robot, she finally snaps into Mama Bear mode as soon as said robot hurts the bird she was trying to protect and utterly demolishes it. This is especially awesome since she started off merely fawning over Sonic being her hero, but with this moment, she finally became a hero in her own right—
Add to that her mere reasoning with another Killer Robot into becoming her friend. Then, in a similar fashion to her 2006 moment, defending it from Sonic. She'll value Sonic over the world, but she cares enough about others to stand up to him.
A mini moment of Awesome, but in E-102 Gamma's story, the bot gets to fight and win against Sonic, having him dead to rights and being mere seconds from blowing his brains out point-blank. Yeah, he's an Anti-Climax Boss, but it's the first, and so far only, time one of Eggman's badniks has managed to defeat Sonic one-on-one; and gives you an idea of how dangerous Gamma may have been had it not been for his heart.
Gamma: Dr. Robotnik... Enemy... Master registration... Deleted... E-Series... Friends... Must save...
Eggman gets a major one, when one realizes he set up a game-spanning Batman Gambit for his scheme to obtain the Chaos Emeralds! Every time:
He sets up a Macguffin Delivery Service by telling Sonic his plans. This results in every single Chaos Emerald being handed to him, because he's anticipated Sonic would get them first to try and stop him:
1 was retrieved simply by playing possum when Sonic and Tails were off-guard.
5 and 6 come to him by not taking chances and simply sending out his own troops to get the Emeralds first, so he's not putting all the cards in Sonic's hands.
He attempts to be Crazy-Prepared with the Egg Carriers, the secret missile to destroy Station Square, and the success of the E-Series. Even if it didn't all work out, it sure shows he's stepped up his game.
Big the Cat has a surprisingly badass final act for being, well, Big the Cat. Firstly, he confronts Chaos 6, a nearly perfect being, fishing out Froggy, who has one of the Chaos Emeralds powering up Chaos at the time and giving him a tail — which gives Sonic and Knuckles the chance to finish the battle — ending with Big using the abandoned Tornado II to escape the Egg Carrier and safely fly it all the way back to his home in the Mystic Ruins, despite never having driven a plane before. And all that just to get back his polliwog pal. That's some friendship.
As hilariously anti-climactic as the fight itself is, Big's encounter with Chaos 6 deserves a bit more credit: he has no idea what's going on and was just trying to get his friend back, basically getting roped into this whole situation without any warning. For all intents and purposes, he's just an ordinary bystander from the Mystic Ruins jungle that far exceeded his comfort zone. But when Froggy's trapped inside that hulking liquid monstrosity, Big doesn't back down; he freaks out for a bit, but he's still determined to save him. Even after Sonic offers to get the frog out himself, he stares Chaos 6 down and promises that they'll get him out of there. And he does. Not even a pissed off water god with six Chaos Emeralds can keep him from his best friend.
After all that fails, Chaos, in his weakest form no less, manages to recollect all of them plus the seventh in mere minutes. No wonder Sonic was absolutely shocked when he got that far.
Sonic running down the side of a massive skyscraper in Speed Highway.
The original Egg Viper face-off was going to be far more heated, with an unused banter between Sonic and Eggman that makes clear they are well and truly done with each other's butting-in, and this was their final, very personal showdown. This conversation was in fact edited into the Sonic X adaptation.
Eggman: How dare you, Sonic! You continue to always, always, always stand in my way! And this time, you destroyed the Egg Carrier, you defeated Chaos, and now you're even leading my base into ruin! In that case, I'll show you the ace in my sleeve. This time you're finished, Sonic!
Sonic:This time, I'll make you realize... that no matter how many times you try, NO WAY! You will always fail!
Even after Sonic beats the Egg Viper, Tikal warns he's not done yet, with Eggman making one last dive at Sonic that, if you're not evasive enough, will destroy the structure you're standing on, taking you down with him.
Sonic becoming Super Sonic. The whole city cheering "SONIC! SONIC!" as Tails explains that it's not only negative forces that make the Chaos Emeralds work, and then we see Sonic going Super for the first time in true 3D. Spine-chilling, and doubly so with the Theme Music Power-Up.
Eggman has a fridge awesome moment: Station Square is destroyed in the end, meaning that Eggman technically won.
Sonic and Tails have a shared ending, with them running around Mystic Ruins together right after being reunited, as their respective Image Songs play. It's both awesome and heartwarming.
alone is just awesome and really gets you excited before you start playing. Open Your Heart really adds an extra layer of epicness to it, as well.