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Music: primarily radio. Try your local community/college station and genres stations. I also go to concerts/dances even if I don’t know the artist/genre.

Movies: big lists (eg AFI top 100), small lists (ie if you liked x check out these), and the indie movie theater calendar.

Although this really isn’t a method of discovering good music and movies but rather of exposing yourself to so much variety that you can’t help but learn about your tastes and the world of art in the process.

Good music: spotify algorithm. Good movies: somewhat unsolved problem for me. Mostly coverage in traditional news formats, movie industry awards, friends, imdb.
I personally use rating aggregation sites like acclaimedmusic, rym and metacritic.