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just to threw some more names in for you op:
>Deathnote, Evangelion, Cowboy Bebop and Fullmetal Alchemist
four very different but very well known anime series. Deathnote is an entry level thriller with rather intelligent story. Evangelion an epic mecha classic with focus on philosophical things, cowboy bebop is all about its unique genre mix between sci-fi/western/jazz-music with exceptional artstyle and the best music in any anime ever while fullmetal alchemist gives a great story and steampunky universe and great action and humor
>if you like realistic/futuristic crime stories
ghost in the shell/ psycho pass
>...absurd over the top humor and action
kill la kill/ gurrenn lagann
>...cute stuff
toradora (classic love story)/ k-on! (moe heavy and perfect to chill)
>...fap material
oreimo(harem) /fate kaleid liner prisma(yuri)
seriously, dont watch these