Right then, must play games...
Megaman 2/X/Zero 2/Powered up/Maverick Hunter, any of those. You need to play at least one great megaman game, although any megaman game will do.
Mario 1/3/World. One of these mario is essential.
Umihara Kawase Shun Second edition: The DS collection is also worth looking at, but you need to play umihara, one of the best physics platformers.
Shadow of the colossus: Amazing game, ICO is also worth looking at, but I think SotC is easier to really enjoy.
Loom: As far as point and clicks go, loom will let you enjoy what makes the genre work, and never need to worry about the pitfalls in the genre for the most part. Steam version is good.
Portal 1/2, either, although I suggest both, they will be the fondly remembered games of this generation, these will be the mario brothers for the 2000s.
SMT: Persona 3/4 either or, I suggest persona 3 portable most though. One of the best JRPGs of all time.
Chrono Trigger: Amazing game, so much is said about it
DQ 4/5/8/9, the others are good, but these are probably the better examples of what the series is.
Street Fighter 2 (preferably HDR): Classic fighter that still holds up just as well as the best of the modern.
Castlevania Symphony of the night: Amazing game, although I prefer Dawn of sorrow.
Super Metroid
Sonic 3 and knuckles: One of the best platformers ever made.
PainKiller: interesting FPS that goes for pure fun.
Bayonetta: The ultimate evolution of DMC, which was for a while, the ultimate evolution of the Street of rage style side scrolling beat em up.
Bangai O, classic amazing shooter.
Sin and Punishment 2: Much improved from the first.
Ikaruga: One of the best shooters made
Gunstar Heroes: Run and gun, contra, but not so evil.
Lemmings: One of the classic puzzle games
Tetris: See lemmings
World of Goo: See Tetris
Links awakening DX: Shows what a handheld can do.
Ocarina of time: Revolution in games
Maljoras mask: Shows how little the engine defines a game.
Mario 64/Galaxy, amazing platformer
There is soo much more I could write, but this will do for now.