Easily Cowboy Bebop. It wasn't the first anime that I watched and enjoyed, but it was the one that showed me how good the medium can be.
Deep, nuanced characters with individual goals and ambitions which were rarely coincident with those of their peers, and which were almost never convenient to follow through on. A rich setting full of people who you could tell had stories all their own outside of the main narrative. A very well paced and carefully delivered main narrative interspersed with smaller, equally compelling stories which drew the watcher in and made them want more. A nearly perfect combination of comedy, action, suspense, and drama which made it feel unscripted and unpredictable (with one notable exception in the form of a one-off episode about old fridge food). The entire show was brimming with style and grace, and it never felt like it was dragging its feet or filling for time.
Then, of course, there was the music. The score was just as eclectic as the show, going from fast-paced guitar riffs to jazzy "travel" music, even finding time to create moving, soulful ballads where they were necessary. The music never felt forced or out of place, as I find to be the case all too often with Japanese scoring.
Following that would have to be Trigun, which would have been first if it weren't for the disappointingly watered-down take from the Maximum manga series. Still a good series, but it absolutely pales in comparison to the source material.
It's just something I don't think I'll ever grow out of! Sailor Moon is what got me into anime, and it was the first one I ever watched. 
