Mine story is actually the reason Zelda is my favortie series as well. When I was 3, in the grand year of 1999, I played A Link to the Past on the Super Nintendo, not only as my first Zelda game, but as my first EVER! Ever sence then, Zelda has been the fore-front of my gaming life.
The first Zelda game I played was Phantom Hourglass. I just had a DS and played Mario games but I was getting tired of playing the same game over and over again and I stumbled upon the Zelda game. I was curious about the world of Zelda because of SSBB where I always had to fight Link first. I wondered who the "Link" was and googled it and then I wanted to play Zelda games. So I found PH and finished it in a week and then I bought TP. I got hooked to this magical world ever since
Mine was Twilight Princess. I had played the Brawl games and wanted to try out Zelda (She's the character I'm best at in those games). I found TP at a game shop and decided to get it. No one was able to get me off the Wii that night, or the next... Or the next. I've been playing Zelda ever since
I'm pretty sure it was Alttp on my old SNES. I remember getting stuck way to much and not getting very far (I must have been 6 or 7 years old) I didn't really get into the series until OoT first released back in 98 I believe for the N64... I was hooked for life.
My first game was The Wind Waker. My sister and I were still rather young when we started it, so we didn't actually get into Zelda until a year later, when we found a walkthrough on the wonder of the Internet and realized how to get through Forsaken Fortress. We've loved Zelda ever since.
I played Ocarina of Time with a friend. At the time I didn't really like Zelda, and didn't think it was cool. But we stayed up all night trying to beat the Shadow Temple, and it was the coolest and scariest thing I had ever seen on the N64.