If you're into storylines, and don't mind them getting very wacky, you should buy the two Trauma Center games, they're both excellent and have all sorts of super drama medical explanations of life, death and storyline-spitting patients.
As well, Dragon Quest Swords has a rather fleshed out storyline for an arcade game mixed with an RPG.
If you don't mind really hard roguelike gameplay, or are willing to go for something extremely bizarre, Baroque is great because it has a gradually revealing plot based on you dying, somewhat like Planescape: Torment and there's quite a bit you need to put together the pieces to understand (the game proper will only present so much of its plot to you).
Disaster: Day of Crisis doesn't have a good storyline, and neither does Resident Evil 4, but if you haven't played either, the action is quite lovely in both and yet they have a lot of storyline detail to keep you occupied. In that vein, if you haven't played the older Resident Evil games, it may be worth it to play through Umbrella Chronicles too, especially if you like lightgun games, because it has a nice story digest of the series in general.
No More Heroes is a prime choice as well, it's both a top-notch action game and its got a warped, fantastic story with awesomely drawn antagonists, great dialogue and a really greatly depicted protagonist.
Of course, Twilight Princess is great, though not that story-oriented, it has more of the flair of a story-oriented game that Zelda always has.
If you like playing old games, Secret of Mana and Phantasy Star IV are excellent choices, even today.