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.
Sonic: Aww, yeah! This is happenin'.
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.
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 in Sonic Adventure, the bot gets to fight and win against Sonic, a mere second from blowing his brains out point blank. Yeah, it'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 gets another one after leaving the Egg Carrier. While descending down to the Mystic Ruins, he starts having "flashbacks" to Beta being rebuilt, the other E-Series bots being sent away, Amy, and finally Robotnik. After the images flash through multiple times before settling on Robotnik, he decides to fully cement his Heel–Face Turn.
Gamma: Dr. Robotnik... Enemy... Master registration... Deleted... E-Series... Friends... Must save...
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 was off guard.
2 was claimed by averting a Death Trap scenario and gassing Sonic and Tails immediately when they didn't see it coming.
He plays off Knuckles' gullibility, which in the long run nets him emeralds 3 and 4.
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, then it 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, giving Sonic and Knuckles the chance to finish the battle, ending with Big managing to use the Tornado II to escape the Egg Carrier despite never having driven a plane before. And all that just to get back his polliwog pal. That's some friendship.
After all that fails, Chaos, in his Zero form no less, manages to recollect all of them plus the seventh in mere minutes.
Sonic running down a massive skyscraper in Speed Highway.
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.