The first anime I watched, like so many other people on PC, was Pokemon. It was probably a close-run contest between that, Yu-Gi-Oh and Digimon, but as they were advertised alongside the other cartoons in the childrens' block of programming, I didn't think anything was really different.
The first TV anime I watched knowing the differences between anime and Western animation - as well as the first one I watched with subs - was FLCL, and I wouldn't recommend this as the first pure anime experience. I imagine I'll come back and appreciate it more in time, but I'd much rather have started with the infinitely more accessibly Cowboy Bebop (which I really, really like). The first cinematic anime I watched was Spirited Away,many years ago, although I might not have watched it with subs for the first time until just last year!