My first anime was Death Parade, and I just fell in love right away. I found the stories to be much more engaging than many of the TV shows I was being exposed to at the time (and since, tbh), along with the visuals being easier to actually see; unlike the direction that many other forms of media seem to taking these days making everything so damn dark you can't see what's going on.
I don't remember exactly which series I started watching beyond that, but I remember Bleach being one of the most memorable. After that, I branched off into a bunch of different genres.
VulcanSphere