Sailor Moon was the first that I caught a glimpse of. Then Cardcaptor Sakura (dubs), but finally the first thing I watched subbed was Seto no Hanayome.
I was casually watching anime for a few years so I don't remember what my firsts were (except Pokemon as I'm not counting that as I didn't know it was anime at the time) but the show that got me to start actively watching it was Angel Beats.
Card Captor Sakura when I was very young, Ghost in the Shell when I was a bit older. I still rewatch them occasionally, and they both hold up pretty well.