I definitely recommend Skyward Sword (it's a modern Zelda game and VERY different from ALttP; it'll give you quite the spectrum as far as Zelda games go), as well as Majora's Mask and Ocarina of Time.
yeah, same for me when I first played Links Awakening. I pretty much I had no idea why I was killing these monsters but I knew if I didn't they'd kill me so i ended up almost beating it, getting frustrated, then retrying about a year later and understanding what was going on and beating it finally.My first Zelda games is Legend of Zelda: A Link to the Past. It was really fun playing without knowing what was the plot. Hehe, good times...
I played about 1/3 through and then kinda stopped but I started again about a year later when I understood it more ( I was like 5 or 6). You should definitely finish LInks awakening.Links Awakening, although I never completed it. The first Zelda game I completed was Phantom Hourglass.
The first Zelda game I ever played was Link's Awakening. It was also the first video game I ever played, It's still one of my favorite games ever. I guess you can see this in my signature too though.