Cecil's fight with his own inner darkness to become a Paladin turns him from a depressed, unsure knight into a confident, walking engine of physical destruction, ready to fight for his beliefs instead of letting himself be forced to do nasty things against his will. Incidentally, the easiest way to win that fight is awesomeness in itself: sit there and take the hits, paying penance for what Cecil did as a Dark Knight.
Probably the biggest moment of awesome in the game is when Cid leaps off the airship riding what is apparently a nuclear bomb, in order to close the entrance to the underworld. After kamikaze'ing himself and then falling hundreds of feet, he survives and walks himself to the dwarf kingdom to recover. Then, while recovering, when he hears about the obstacles you face with your shiny new airship, which can't cross lava and needs a drill to get back to the surface, he gets out of bed and still has the strength to apply the necessary upgrades! Truly a mark of Badass.
Before his final defeat he deviates from many villains by appearing with the other elemental lords and telling you that (unlike other villains who merely try to outnumber you in an ambush) they have learned the power of team work from your past victories and proceed to fight you as a team.
Due to technical limitations, they fight you one at a time because having four bosses on the screen would be impossible even in the DS version. The spirit of the dialog clearly indicates that they're fighting together, even if the game doesn't show it. That said, you are given no chance to save or heal outside of battle until all four are defeated at once.
The assault on the Giant of Babil. The entire armies of the world, with every bit of technology and magic at their aid, assault the Big Bad's ultimate weapon. So. Very. Awesome.
Edge, speaking to Rubicante in the Tower of Babil, especially in the DS version. As Rubicante gives a pitying speech about how mankind is a slave to their emotions and can never know true strength, Edge instantly slips into a Unstoppable Rage and delivers an awesome line to Rubicante amid a swirling storm of lightning.
Edge: You think our rage... a weakness? Then let me show you HOW WRONG YOU ARE!!
The two instances the "Theme of Final Fantasy" plays. It's not Awesome Music for nothing.
The summoning of the Lunar Whale. The awesome music accompanying the scene is nothing short of epic.
Cecil calls out to Golbez in the finale, calling him his brother. In the original, Golbez merely turns around to say farewell; in the DS version, we see him as Theodore as he says goodbye. It's much more effective if you're used to the original version.
After Cecil's inexplicable "Stay in the Kitchen" abandonment of Rosa and Rydia on Earth before the final battle, they show up once he lands on the moon, having apparently been sitting in the entrance chute of the Lunar Whale during its trip through space. Utterly ridiculous, but nobody cares.