I watched pokémon and dragonball z when I was young, but my first real anime was either Madoka Magica:
or Shugo Chara:
Which has been probably more than 6 years ago since I watched that, it's so old it wasn't even on my anime list saying that I completed it. My friend who really loved anime introduced me with "easy to watch" animes like these. And ever since I've been part of the culture.
I actually have no idea what my first was. I didn't even know what anime was at the time, I just recall watching this show that was like a cartoon, but cooler, more detailed, more mature.
I specifically remember a scene where there was some weird giant creature that was helping out someone participating in a sort of race by fighting two other giants that guarded some gate, and this helping giant ended up getting torn apart by the duo and blew up, leaving no trace of itself or the guards.
My other memories of it are too vague to really describe. But they always remain somewhere in the back of my mind, irking me. I wish I could find the show and watch it again, so I can lay it to rest.