Was it The Walking Dead or something else? For me, I was about 12 years old when I first saw The Omega Man with Charleton Heston. He's the lone survivor of a plague that wipes out the entire population, except for a few people who are still alive but physically changed by the effects. They technically weren't zombies but they were scary (to a 12 year old girl!) and dangerous. That has always been one of my favorite movies, and is still tons better than the Will Smith remake I Am Legend.
How did you get introduced to zombies?