I'd start with those games;

Donkey Kong (Arcade), because the arcade game made Nintendo what it is today.
Super Mario 64 (Nintendo 64), one of the Nintendo 64's first ever games, my cousin introduced me to it and I loved it as a kid.
DOOM 64 (Nintendo 64), a port to the N64, it is the grandfather of FPS.
Goldeneye 007 (Nintendo 64), the father of FPS, and it started the era of the modern FPS you usually see from AAA publishers nowadays.
Metal Gear Solid (Playstation), if you had a Playstation back then, then you have probably heard of this game, which is considered one of the greatest games of all-time.
Any of the Spyro the Dragon games on Playstation, I actually never played them myself, but it's undoubtedly a classic to play.
The Crash Bandicoot games on Playstation, again, I never played them, but it's also a cult classic.
The Legend of Zelda: Ocarina of Time (Nintendo 64), one of the greatest games of all-time, also defined the N64 too.
Perfect Dark (Nintendo 64), the spiritual successor to Goldeneye 007. A very good game to play, that also pushed the N64 to its limits too, as the N64 was at its EOL.
F-ZERO GX (GameCube), you don't usually see this game recommend to anyone looking to play some retro games, but this, my opinion, is the best F-Zero game in the franchise. It's very difficult, but once you get used to it, it's rewarding.

The most important part about playing, is to have fun! I hope you enjoy the games you play!