Gosh, there are so many games that I could list that actually mattered for introducing me into a genre of games. I'll try my best though to limit it to 5.
Mega Man 3
I remember working with my Grandfather at flea markets and would earn $5 a day. I saved up my money and bought this. I was so amazed when I played this game that I saved up my money and eventually bought every Mega Man game I could find (including Mega Man X). This game introduced me into the awesome aspect of 2D side scroller where you got to fight avatar-like bosses, and upon beating them, gaining a portion of their powers. For the NES it was gold. This genre would lead to me an appreciation of the 2D platforming of Super Mario Bros., Sonic the Hedgehog, Teenage Mutant Ninja Turtles (games with co-op were awesome), Streets of Rage, Golden Axe, etc.
Super Mario RPG: Legend of the Seven Stars
This game really turned me on to the RPG genre. I never even knew what a RPG was until I played this game. It took me a moment to realize that it wasn't going to be all about action, and a lot about reading and planning. But once I gave it a chance, I was hooked. I ended up starting this game over 3 times, because my mom returned it to BB and re-checked a different cartridge :( This would be the game that would lead me into other epic games such as Chrono Trigger, The Final Fantasy Series, The Tales Games, and the Grandia games (story wasn't that awesome, but battle system was epic). It is also why I can appreciate MMORPGS.
This game introduced me into the world of FPS games. Even now, I consider this one of the best FPS games to date. It introduced a great FPS experience, a mature storyline, and multiple reason to replay it. This game is the reason I would eventually play Half-Life, Halo, TimeSplitters, Left 4 Dead, etc.
Super Mario 64
While I did play and beat the NES and SNES Super Mario Bros. games, it wasn't until this game that I developed a new appreciation for the Action/Adventure genre of 3D games. This game was revolutionary by most standards for it's time. It was because of this game that I got into Sonic Adventure, Legend of Zelda: OOT, Banjo-Kazooie, Nights into Dreams, etc.
Although I consider the RTS genre probably the hardest and rarest genre, it was this game that introduced me to this genre. My best friend recommended it and I gave it a try and I couldn't help but feel that it pushed my multi-tasking abilities and strategic abilities to it's limit. There is nothing more rewarding than defeating an opponent because of a superior strategy. It was because of this game that I would play it's expansion Broodwars, the Warcraft genre, etc.
Well this the list I compiled, I do feel I had to leave things out that were also important, such as Sonic R introducing me to the Racing Genre (although it is probably my least favorite genre). But the important genres were covered.
Strategy/2D Real-Time Strategy