The first anime I ever watched was definitely Sailor Moon, the DiC dub, which I still have massive nostalgia for. My sister used to watch it and I would watch along, even though it was "girly". Rewatching it now (with subtitles this time), I'm amazed at how much fanservice is in it. Guess I know now why the transformation sequences made me feel funny.
From there it was Dragon Ball Z on Toonami, but I can't remember if that came before or after Pokemon. I had no idea Sailor Moon and Dragon Ball Z were from another country at first. I thought Dragon Ball Z was a Warner Bros. show for some reason, I guess because the eyes reminded me of Looney Tunes or something. It had a massive impact on me though because it was the first cartoon I ever saw where they actually animated people fighting, instead of just having a dust cloud with goofy sound effects. I definitely knew Pokemon was Japanese because I thought some of the designs (especially Lapras, for some reason) looked "Japanesey"; I guess that was my way of identifying the overall anime aesthetic of the time.
After Dragon Ball Z ended and I had watched a few other Toonami-aired shows like Outlaw Star, and around when Naruto started to air, I stopped watching anime since I couldn't identify with the newer shows, and I thought I had "outgrown" it. I had no idea the massive volume of anime that existed outside of the few that aired on the cartoon channels on American cable.
It wasn't until the summer after I graduated high school, when I dated a girl that was into anime, that I intentionally sought out anime again. I watched all of Neon Genesis Evangelion on some streaming site and absolutely loved it. One of my roommates in college was a massive weeb, and he introduced me to Toward the Terra (the remake), and I loved that too. From there I decided to watch more anime, and started lurking /a/. Haruhi was my first slice of life anime, and JoJo had just started to air, so I got into that as well.
tl;dr, first became conscious of "anime" during the Pokemon/DBZ era, first anime I actively sought out was Evangelion