Bond escaping from the chemical weapons plant on a motorcycle, which he then rode off a cliff and caught up with a plane.
At the beginning when Bond jumps from the top of the dam. The free-fall and eventual landing scene that follows is several seconds, maybe even up to a minute, of a single sound of whooshed air. When this movie first came out, that was the moment when people knew this Bond film would revive the series.
For me, the scene when Pierce Brosnan became Bond was when he and Natalya were facing the Russian official who threatened to execute them:
James Bond: What, no small talk? No ... chit-chat? That's the trouble with the world these days, nobody takes the time to perform a really sinister interrogation. It's a lost art.
Also, the final fight between Bond and Janus. Competing for the title of best fight in the series with the fight between Bond and Red Grant in From Russia With Love.
According to the DVD commentary, it was scripted and filmed to be a homage to the legendary Bond/Grant fight.
Bond barely acknowledging that he is fired upon when setting bombs in Janus' base, just giving a slight flinch when bullets hit his cover.
Natalya, not Bond, is the one who ultimately foils Janus's scheme by reprogramming the sattelite to drop out of orbit. Made even sweeter by Boris taunting smugly that she can't stop GoldenEye from firing: "She works on the guidance system." Yes, and a lot of good being able to fire the thing will do you when it's burning up over the Atlantic, won't it?
Natalya also gets a few other awesome moments, such as when - on the run after barely escaping Severnaya with her life - she walks into a computer store with nothing but sheer audacity and bluffs her way to net access by posing as a buyer from a university.
Natalya beating the crap out of Boris when she meets him in Janus's control room. It took two mooks simply to restrain her. And judging from the cut to Janus's smirk, even he thought it was pretty awesome.
And her having the presence of mind while terrified by the office being shot up to fake an Air Vent Escape.
The title sequence, both because it was a new take on the classic Bond opening that set the scene for those in later films, and the powerful imagery with the Communist iconography being destroyed.
For a slightly more obvious Moment of Awesome, see no further than the wonderful applications of Tank Goodness.
The grenade launcher in Surface and the remote mines in Facility are very fun to play with, especially in multiplayer.
At the end of the trainyard mission, Bond steps onto Alec's train, and two guards appear on either side of him. He guns down the one on the left, whips around to kill the one on the right, and then pauses, turns, and adjusts his tie right as the doors slide shut. Classic Bond.