Technically the first anime I watched was Pokemon. As a child though, I considered it just a cartoon and didn't really know what anime was.
In fact, I hated anime up until about age 12...I watched the movie Spirited Away and was captivated by how beautiful the visuals and storytelling was, and wanted to see more media from Japan. Still one of my favorite movies of all time. Many of my friends at the time recommended I watch Naruto (this was around 2006/2007 when the series was going strong), so that was really the anime I consider to be my first, and had the most impact on my life from that point on. Funny thing is, I always preferred shoujo anime back then, but after rewatching Naruto a few months ago, I enjoyed it a lot more as an adult. I used to dislike the character of Naruto as I found him annoying, but watching through the whole series I can honestly say he's an amazing character and I admired how he persisted even though everyone thought he was a loser and would never become anyone special.