Easy enough, what games are your "Firsts" for certain things?
Here's mine as an example.

First Game Played : God knows, but I remember it being a tape and taking half a century to load.

First Big Multiplayer Game : Halo 3 or Runescape

First Game to convince me that they were more than just toys : Metal Gear Solid

First Game to make me cry : Metal Gear Solid again.

First Game to make me proud of beating it : Tie, Either FF7 or Ninja Gaiden Black

First Game to teach me something complex : Tie, Halo 3 and PGR, Used both in a college essay on emergent behaviour.

Edit: Just realised I posted this in Off-Topic, would someone with magical mod powers please move it. Much Obliged :)

First game played : probably something on tape as well.

First system owned : Acorn Archimedes 2000.

First game owned : Errr... "owned" Elite, Swiv and E-Type Jaguar Racing. Really owned, Street Fighter II Championship Edition.

First game finished : Streets of Rage II - in the days when "2 player" was synonymous with "local multiplayer co-op"

First big multiplayer game : Does GoldenEye not count? Battlefield 2.

First online multiplayer : X-Wing Alliance.

First MMO : Final Fantasy XI.

First Game to convince me that they were more than just toys : None - disagree with me and I'll disagree with your definition of "toy".

First Game to make me cry : Still waiting. I don't get that emotionally invested in something I can switch off.

First Game to make me proud of beating it : Probably Streets of Rage II, I was probably only 10 or something.

First Game to teach me something complex : Still waiting. The more I think about it, the more I'm scared of what prejudices and lies gaming has filled my head with.

First game played: Croc on ps1 (as far as I can remember)
First big multiplayer: CoD4
First game to convince me they were more than just toys: Same as OP, MGS
First game to make me cry: FFX
First game I was proud to beat: Can't remember, probably FF8

can't think of a game that taught me something complex, will think about it some more.

First game played: LEGO Racers (PC) (and I now own pretty much every single game with the LEGO label on it)
First game beaten 100%: Croc 2 (PC)
First console owned besides PC: Nintendo DS
First Steam game bought: Left 4 Dead 2
First Final Fantasy: IV/II
First first-person shooter: I honestly don't remember but I think it was Portal, if that counts as a shooter...
First serious horror game: Amnesia: The Dark Descent. Still too scared to go through the prison
First major multiplayer game: Starcraft, though I only played SC2's multiplayer
First game that taught me something about myself: Mass Effect. I have trouble even taking its Renegade options - I'm not a morally bankrupt enough person to even consider going any further down the scale than "neutral good".
First 'Battlefield moment': Running down an alley, a tank drives past...with its main gun pointing straight down the alley at me. Then I respawn and a different tank nose-dives off a hill. I had to consciously stop myself saying "Holy shit!"

First FPS: Some James Bond movie thing I don't even remember.
First RTS: Either Starcraft or Age Of Empires.
First RPG: Not even the slightest clue.
First Driving: Gran Turismo or Driver.
First Fighting: Tekken. First and only. Except Blaz Blue, but that's just a thing on the side.
First Platformer: Spyro the Dragon.

First MMO: Runescape, of course.

First Current-Gen game: Resistance and inFamous, same time.
First game I completed: Spyro the Dragon.
First game to 100%: Nope. I'm going for 100% on Dark Souls though.
First horror game: Penumbra, or Half Life if it counts.

First OMG Moment (Singleplayer): Completely pwning NYC in Prototype.
First OMG Moment (Multiplayer): Epic Infected combos in L4D2, or epic intel caps in TF2.

First Game Played: I remember it being on a old brick of a computer and something about a green alien needing to jump about places, called Abe I think.

First Big Multiplayer Game: Call of Duty - World at War

First Game to convince me that they were more than just toys: Grand Theft Auto 2

First Game to make me cry: Heavy Rain

First Game to make me proud of beating it: Gex Deep Cover Geko

First Game to teach me something complex: Silent Hill for the PS1. A game doesn't need to tell you where to go (sometimes) if your willing to explore.

First Game Played: I have no idea, probably the Aladdin game on the Megadrive.

First Big Multiplayer Game: Halo 2, in which I learned that being female is apparently `A Big Deal`.

First Game to convince me that they were more than just toys: I don't think I ever regarded games as `just toys`, but it was probably my obsession with getting 100% gems in Spyro. I took that shit seriously.

First Game to make me cry: I dont think any have. Oh, no I teared up at the end of Portal 2.

First Game to make me proud of beating it: I'm proud of beating all games I beat, but probably the Jak trilogy, not cause it was hard, I just wanted to be able to say I had beaten all of them.

First Game to teach me something complex: Assassin's Creed. Taught me that you have to put up with annoying peasants shoving you because flossing their brain with your hidden blade is apparently rude. Such a complicated choice. To stab, or not to stab.

First MMO WoW. Still playing.

First Game Played : Duck Hunt

First Big Multiplayer Game : Star Trek: Bridge Commander

First Game to make me cry : Mass Effect

First Game to teach me something complex : Um... Battlefield 1942? it helped me pass a WWII test in high school school.

First Game Played : Super Mario Bros./Duck Hunt

First Big Multiplayer Game : Call of Duty: World at War

First Game to convince me that they were more than just toys : Onimusha

First Game to make me cry : Sonic 3 & Knuckles (out of sheer awesomeness, I assure you)

First Game to make me proud of beating it : The Lion King

First Game to teach me something complex : Pokemon Red version

First Game Played: I don't know. I'll go with Super Mario Bros but it might have been something else. I can't remember.

First Big Multiplayer Game: Define "big". Does 32 players count? If so, then Half-Life. I haven't exactly played it with 31 more people but the potential was there. I think the most I've done is 20-ish multiplayer on HL or Opposing Force. Then came CS.

First Game to convince me that they were more than just toys: Blood 2: the Chosen and Delta Force. The former was the first game I played on a PC the latter was the first gmae I see people play in LAN. Also the first I played in LAN. Before then I didn't think of games much more than I think of Tetris.

First Game to make me cry: I don't think there was but some did really make me feel bad. The first one was Unreal 2. God, I hated that ending. I mean in a good way.

First Game to make me proud of beating it: Red Faction. It was the first game I played from start till finish with no cheat codes. Until then I hadn't played that many games from beginning till the end. I either found them boring and quit, or just typed some cheat codes and had fun.

First Game to teach me something complex: Legacy of Kain. Soul Reaver 2 was the first game of the series I played (the demo captivated me - must have played it at least two dozen times). I always liked time-travel and such - LoK is filled to the brim with time travel and the shenanigans it can cause. That made me look up the grandfather paradox (what happens if you go back and kill your grandfather) and generally really taught me a lot about it and the different theories/options/strengths/weaknesses to time-travel variants. I know it's probably not that useful a knowledge but it was complex. And I liked it.

First MMO: World of Warcraft. I wasn't really interested in MMOs beforehand. I'm not interested in them after it.

First Game Played : Super mario world

First Big Multiplayer Game : WoW / Halo 3

First Game to convince me that they were more than just toys : Bioshock

First Game to make me cry : Probably Portal, tears of joy over how fun it was.

First Game to make me proud of beating it : Looking back and realizing I beat Legend of Zelda: Ocarina of Time without a guide

First Game to teach me something complex : Halo 3 multiplayer thought me to think fast and on my feet.

Danyal:
This makes me feel... old?! NO! That's a first too, I think.
Come on, first multiplayer COD4? First FPS Portal?! First everything Bioshock?!

Not everyone was born at the same time as you. Nor did everyone get really into gaming at the same time as you, via the same genres as you. Besides, even though "you don't feel old", criticising us for playing newer games as entry points to a genre sounds a lot like "Damn kids, get off my lawn!" Except it's not "your" lawn. It's a public park that everyone uses in their own unique way. Some kids climb trees, and some like to stick to the swings and never use anything else. And then there are old people who sit on benches, watch the new people having fun and ramble about how back in their day all they had was a stick and some string.

And besides, saying that Portal was my first shooter was not in any way criticising older games like Doom or Duke Nukem 3D.

First Game Played: Yoshi Story
First Big Multiplayer Game: Combat Arms
First Game to convince me that they were more than just toys: Ocarina of Time
First Game to make me cry: Chrono Trigger
First Game to make me proud of beating it: Megaman 2
First Game to teach me something complex: Eve Online. Before then I didn't really understand much about economies, now I'm a great banker.(at least in game)

Vault101:

JemJar:

First Game to make me cry : Still waiting. I don't get that emotionally invested in something I can switch off.

.

you can "swtich off" a movie...or close a book....Im not seeing the distinction there

I'm not making that distinction.

First Game Played : B-Wings

First Big Multiplayer Game : MMO - Ragnarok Online
Lan - Counter-Strike

First Game to convince me that they were more than just toys : Half-Life 2

First Game to make me cry : Finally after all these years, Mass Effect 3

First Game to make me proud of beating it : Alpha Protocol (Only because its the worst game that I've ever beaten from start to finish.)

First Game to teach me something complex : Half-Life 2 (Though me the effectiveness of layered and subtle game design.)

First game to make me quit after the first hour and never play again : TES IV Oblivion

First game to make me rage quit : The Witcher 2

First game played: Pfff... Like heck do I remember that, I think the earliest one I remember was a Batman game of some description (That used to take 30 minutes to get running)

First game personally owned (I shared games/consoles with my brother): Toy Story 2: Buzz Lightyear to the Rescue

First MMO: World of Warcraft

First RPG that made me interested in more than just combat: Mass Effect 2

First game to make me lose track of time: Skyrim

First popular game I played before it became popular: Portal (That game is great if you play it without all the hype about it)

First game that made me practice to get better at: Team Fortress 2

First game to make me dream about said game: City of Heroes/Villains

First game to make me nostalgic: The Legend of Zelda: Ocarina of Time

I don't think I can remember any more firsts... Ahh well, I'm looking forward to some more firsts.

Stealing OPs list, since I'm lame.

First Game Played : Probably something simple, like Tetris or a computer pin ball game I remember playing when I was 4-5.

First Big Multiplayer Game : Counter-Strike. Before that I only played single player games on my Playstation.

First Game to convince me that they were more than just toys : Hasn't happened.

First Game to make me cry : Hasn't happened.

First Game to make me proud of beating it : Hm... Would probably be Spyro the Dragon.

First Game to teach me something complex : No particular one that I can think of, though all my gaming as a child taught me English. That's gotta count for something.

First game finished: Sly 2

First game to make me cry: GTA: San Andreas

First multiplayer game: Team Fortress 2

First Gaming Memories: Watching my father play Wing Commander and Tombraider.

First Game Played: Hard to say since my memories here are shrouded by silly children games borrowed fromt the library. One of the first games I consciously remember playing was called 3-D Ultra Pinball. I still own that one, since I've stolen it from my father at some point.

First RTS: Either Age of Empires 2 or Settlers 3.

First Shooter: I don't remember the name, but I remember that the game was so bad that you could one-shot enemies with a sniper rifle by hitting the barrels of their rifles.

First RPG: Icewind Dale

First Big Multiplayer Game: World in Conflict

First Game to make me cry: The closest I ever came to this would have to be the Mass Effect series in general, which is the series I've become the most emotionally invested in.

First Game to convince me that they were more than just toys: Probably also Mass Effect ... No, scratch that. As strange as it may sound, I'd have to say Call of Duty 4. Not because it was anywhere close to art, but because it had some scenes that really showed me the potential games have as a medium.

First Game to make me proud of beating it : Titan Quest

First Game I nearly became addicted to: Titan Quest

First Game to teach me something complex : I'd say that every game with more complex mechanics than "Go there, shoot that guy in the face" teaches you something complex. But the first game that I really had to put work into to learn was World in Conflict. It taught me to macro- and micromanage, to cooperate as a team and to react and strategise quickly. All that I also knew before, but WiC taught me that I had to do all of them at the same time to be successfull.

First game played: Think it was Star Wars Jedi Knight: Dark Forces II. It was my brother's. We never got beyond the third level. Epic game though.

First game completed: Lego Alpha Team *snigger*. It was really hard as well!

First multiplayer game: Split-screen on Tony Hawk's Pro Skater 2. Oh, the memories.

First console: An Xbox. I was late to the party.

First game of this generation: GTA IV. What a way to begin.

First Game to teach me something complex: Assassin's Creed 2 I suppose. For teaching me about history.

First Actual Game: Those old Putt-Putt games, honestly.
First FPS: Delta Force 2
First MMO: Runescape. Yep. That piece of sh-.
First RTS: Chaos Island, followed by SW: Galactic Battlegrounds.
First Fighting Game: That was some old Digimon game that I played at a friend's house. Awesome game honestly.
First Flight Sim: Fighter's Anthology. Old Jane's Survey sim... Love it.
First Game I Beat: Chaos Island.
First Game I Cheated in: Not sure... Jedi Knight: Jedi Outcast or SW: Galactic Battlegrounds. Forcebuild, oh how I love you.
First Steam Game: Half-Life 2: Episodic.
First Game on the Steam Platform (PacSteam): Half-Life 2 and GMod.
First Space Sim: Wing Commander IV, followed by Prophecy, then FreeSpace: The Great War.