The earliest moment for me would be the lock-on technology for Sonic & Knuckles. Sure, we only had the physicals for Sonic 1 and 2 to attach it with, though the awesomeness kicked up tenfold when I borrowed a friend's copy of Sonic 3 and really got to see its capabilities. It's a shame that maybe up until Vib Ribbon, that kind of technology won't be seen again.
And of course, I can't avoid threads like this without bringing up Crash Bandicoot. When I saw it in action for the first time (from the same friend that I borrowed Sonic 3 from, no less ), I had absolutely no idea that a platformer would end up go into the fresh dimension that it took. I wondered what it would be like in the past if a standard scrolling platformer had a character that could occasionaly walk into the background or closer to the player, but the clever design for the first game really took the cake. I feel honored to see it be my first experience of the PlayStation
And while we're on a PS1 tangent, let's bring up the Dual Shock controller. N64 players may have already had their equivalent and had fond memories of it, but I never really got used to analog control until my brother invested on one some point in early 1999. It took some learning to try and get used to it, as well as the rumble in it whenever I'd take damage or react with the environment someway or other, but it really brought fresh life to the games that were compatible with said controller. Just as well that I had to learn to Dual Shock, especially since I'd eventually be playing Ape Escape later that year, with its complete dependency on said controller.
Yep, call me a 90s kid, but I feel at home with those memories