My first was the original Zelda on the NES, when it came out. My neighbor's dad bought him the game so I got to play it shortly after its release. It was such a different kind of game compared to most other we had played until that point. The freedom to go anywhere and all those secrets. All those items that allowed so many new features as you got further into the game. Like the ability to burn bushes to reveal secret locations and whatnot. It was such a unique and mysterious game for its time and I got hooked to the series instantly.

My first was Twoilight Princess and after that I tried older ones. Really like Link to the Past.

You should chat up S.Peelman. He adores Link's Awakening.

@OP: I definitely played the first two games on NES, but the first one I beat was A Link to the Past. On a side note, this series means more to me than any other, and I can't imagine playing games without it.

Zelda 1 on NES
I barely remembered playing it though

My first Zelda was Ocarina of Time on Gamecube on the Master Quest edition.

Mine was Links Awakening. Which is possibly one of the best games in the series as well. That game made me a big Zelda fan and was the start of my love for Nintendo.

Skyward Sword.

Ocarina of Time on N64. I watched my brother play it initially and was in awe pretty much. Didn't play it myself til about a year later though.

ALTTP I think, second was OoT. :)

Ocarina of Time for me. Then I played Majora's Mask, Link to the Past, Phantom Hourglass and Wind water HD. All of them great, but Ocarina is still the peak for me.