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!