Sailor Moon, in the mid 90s, was the first I saw thanks to it being localized on TV.
First non-localized anime I saw was Appleseed, I was 9 years old then, got it on VHS because one of my dads co-workers thought we'd like it or something so he gave it to him. No idea where that guy got it but I guess maybe he was one of those super rare Otaku from before the internet made anime huge. Naturally it was dubbed in English which was a pain because I'd been learning English for a few months at that point. But the pew pew and mecha was cool as all hell
First full series I downloaded was Hellsing in the early 2000s, probably 2004. Found out about it from people playing Ragnarok, as well as dozens of other shows people were into at the time because all the anime nerds were playing it and using the various boards to connect and discuss their weebery.
I'd downloaded some random Dragonball Z stuff before I got into Hellsing because my Asian friends got me into the manga. But there were only random episodes, AMVs and movies on the peer to peer network I was using at the time though.