These aren't in any particular order, just so you know.
1. Battlefield: Bad Company 2
Between the classes, the graphics, the available weaponry, the destructible environments, the vehicles and the squads, it is a great example of a variety of FPS concepts.
2. Dark Cloud
A fantastic RPG with quick to learn mechanics with a nice gameplay progresion (weapons, characters, towns, enemies, etc). It is a good introduction for the various advantages of ranged and melee combat in RPGs, the art style doesn't require fancy graphics, meaning it still looks fantastic despite being an early-era PS2 game, it introduces the concepts of elements, has survival mechanics that are easy to learn, yet spell horror if not adhered to (breaking a weapon is a horrible experience). It has the basics of every mechanic introduced fairly early, meaning it isn't one of those "10 hours in is when it really gets good" games. Overall it is offers a wide-variety of concepts, at an easy to learn pace while still managing to offer a fair deal of depth and strategy.
3. Golden Sun
Another RPG, this game introduces turn-based combat and multiple character management, setting a decent foundation for any later gaming-ventures someone may take into party-RPGs or RTSs. Also, it can be played well without an in-depth knowledge, but greatly benefits from experimentation and thoroughness, making it a good choice for long-time gamers and new-comers alike.
4. MarioKart (newer is probably better)
MarioKart.
5. ... Okay, let me think. I have FPS, RPG, Racing, an RPG that includes a tangent to RTS tactics because an RTS isn't exactly a genre that everyone should play, given the tendency for a horrifying difficulty curve ... ... what genre have I missed??? Platformer! That's what I missed! InFamous.
I know it isn't a platformer, but it is a Sandbox (which I missed), and it has climby-jumpiness, so it is sort of platformer-ish. Either way, a good game that introduces the Karma system, the third-person shooter, the 'skill screen' for buying abilities, the joy of side-missions, etc. So it does cover a lot of the bases I missed in some small way, and it is a good game in its own right (also a great example of what a super-hero game should be).