Best ones to start with would be "A Link To The Past" for 2D games, and "Ocarina of Time" for the 3D series. While everyone has their favorites in the series (big fan of "Majora's Mask" here), these two basically cemented the tropes and gameplay the entire series uses in each of it's respective forms.
Any one for a DS or Wii (not the Wii U)? That's all the Nintendo systems I have (I'm a little against Mario and for some time against the company, though I broke that as I shouldn't blame the company for my dislike of Mario (Sonic fan)).
Slim pickins' for the DS: you have Phantom Hourglass and Spirit Tracks. IMO, they're okay, but not great.
For Wii, there's Twilight Princess and Skyward Sword. I really enjoyed both of those, though popular opinion seems to be kinda divided on them. Fortunately, there also exists the ability to play Gamecube games on the Wii, so you can also play Ocarina of Time if you can find that disc. In fact, that disc SHOULD also have Master Quest as well (which is just Ocarina of Time, except with dungeons remixed to be crazy hard). And glory be, Ocarina of Time, Majora's Mask, and Link to the Past are all available for download on Virtual Console!