First became familiar of Japanese animation with these:
Robotech
Voltron
Belle & Sebastian
Grimm's Fairy Tale Classics
Even with "American/Western" cartoons, quite a few were animated in Japan during the 80s, and in some cases, you see the Japanese quirks in character expressions(kinda the closest of having genuine American/Western anime, with the exception of IGPX).
But, I became a real LOVER of anime after watching THESE three movies:
Robot Carnival (we seriously need a Region 1 release for this one, dammit!)
Vampire Hunter D
Windaria
The latter two made me fall in love with Yoshitaka Amano and Mutsumi Inomata, before I became aware of their involvement with Final Fantasy and the Tales series. And these three movies were heavy inspiration on me as an artist and even a storyteller(all three have worlds that are shrouded in wonder and mystery where not every aspect is spelled out and even when you get a firm idea of the worlds, you still feel there is more bizarre mystery to them).
Basically, all those got me ready by the early 90s to the point when the ORIGINAL Dragon Ball(yes, pre-DBZ becoming a pop thing), Teknoman, Saban's various localizations of Japanese animation(Gulliver's Travels, The Little Mermaid, Around the World in 80 Dreams), then eventually DBZ, Sailor Moon, Ronan Warriors and such were all welcomed by me with open arms!