Furthest back that I can remember are Doraemon both films and series from around 80s 90s and some other tokusatsu. All on VHS. Can not remember which one exactly is “the first”. I start early ;)

Then some series that I never watched from start to finish on cable network. Like Ranma, Sailor Moon, Oh! My Goddess, You’re Under Arrest!, Houshin Engi, El Hazard, CC Sakura, Galaxy Angel. I always try to be in time for Sakura and Galaxy Angel airing, because cute girls. Never did, because school.

edit to include Utena… it became a source of running joke between me and a girl next door.

Some certainty with title that marks the moment I start actively seeking out and watching more anime. Probably Sister Princess, fan-subbed, burned on CD-R. Good times.