God, this is a hard question to answer. There are a lot of moments that stick out to me as really "wowing" me at the time.
In all honesty, there was so much I missed in my first playthrough. I missed the Prehistoric Hunting Grounds entirely, missed one or two sidequests, and spent literal hours at the Millenial Fair spamming the games to rack up SP.
The part that probably stands out to me the most, however, is the events surrounding Arris Dome. Traveling down into the basement, catching the rat (which took me forever), and fighting the Guardian boss (which also took multiple attempts) were really awesome.
I remember my excitement in having defeating the Guardian (finally), then traveling into the final room only to find a corpse and boxes, one of them broken open. I specifically remember the fog in the air and a spinning air conditioner-type fan against the back wall. When I found the seed and then returned it to Doan, I was excited beyond all reason.
You know how Arris Dome has sort of an "S" shape path before you reach the main area with the inhabitants of the dome? I went to the part in the middle, just below Doan and the others. I remember pushed the up button so Crono ran against the invisible wall, and he sort of shook his head no like he was giving an inspired speech. I ended up doing just that; out loud, I gave the people of Arris Dome a homemade speech about life and hope and, in my eleven or twelve year old mind, thought I was hot shit.
Good times!