The three you NEED: Link's Awakening DX (not DS, DX), Wind Waker, and Ocarina of Time- probably Master Quest.
The story gets really important, especially in Ocarina. There's a lot of revelation in the end, and mystery. Characters are deep... You'll feel like you really know some of them well. The graphics in OoT are stellar. For its time, that is. But even playing it now (I first purchased it last Christmas for GCN after having not played it for several years) I am still impressed. Ganondorf is frightening, and Link looks like a real hero, and Zelda looks like a real princess... They're just fantastic.
Zelda, especially OoT and WW (Wind Waker, if you didn't know) are quite story intensive, particularly for adventure games. The cinematics, most notably in these two, are great. The movie sequences are fantastic, not too long, captivating, and masterfully painted (pixellated, rather). The gameplay graphics match up to the cinematic graphics.
Ocarina opens up with Link, a Kokiri forest boy who hasn't got his fairy yet, being woken up by a fairy finally. Link is told to go to the Deku Tree, guardian of the Kokiri Forest. Link battles through the dungeons in groups: the first three, the last five, then Ganondorf's Castle. It is perhaps one of the greatest games ever made, if not the greatest. Check the GS review if you don't believe the "hype".
WInd Waker: Don't let the untraditional cel shading put you off. The cel shading is the best of any game so far, and the gameplay remains the same. The graphic style goes well with the world. In the story, Link's sister is kidnapped on his birthday and he tries to go save her. The plot is much, much more complicated than this, but that's all I should say- spoilers.
Both of these games have epic musical scores- and I'm a violinist. I can truly say, the violin synthesizing in WW is QUITE good, and I can always tell if the string instruments are real or not... I could tell in WW, but they were very, very good, and the music was a long enough loop to not sound like a loop most of the time. OoT also has astounding sound. I particularly liked the Gerudo's Fortress theme (I guess my parents listen to too much Govi...), but it was all great.
Link's Awakening is not a typical Zelda game, but it holds up well, nevertheless. You wake up on a mysterious island that has recently been plagued by monsters. You meet a lot of characters as you travel through eight dungeons, trying to wake the Wind Fish by gathering the Instruments of the Sirens so you can go home. The gameplay is simple to master, there's tons of items, as in any Zelda game, and it's easy enough to hit the enemies without going too close (unlike A Link to the Past).
Zelda is an epic franchise. Most of the games are pretty cheap nowadays, and new ones are coming soon (Minish Cap, Zelda 2005...). It's a completely worthwhile investment. At least rent one.