My first one that I watched on television was I think Bobobobobobobo (Yeah... great right) but my first one that I watched the full thing of was Angel Beats.
It was Naruto...I was really nervous because it was my first time and I was like OMG so nervous desu but he walked me through everything and now i'm in LOVE with the Nippon DX
I watched TONS of pokemon when I was younger, it was my absolute favourite TV show at the time (and I loved it even more because I played the games and collected the cards) However I don't know if that really counts because for that (along with sailor moon (another childhood favourite show of mine <3) I didn't know that they were an anime. The first one that I watched that I knew was an anime was probably Death Note, I had been recomended it by a friend and I totally loved it when I watched it for the first time.