Anything by Studio Ghibli is good. My personal favorites are LaPuta, Castle in the Sky and Spirited Away. These are both fantasy.
I prefer the older anime classics
Galaxy Express 999 ( really old 1970s anime-you might be turned off by the character designs but the story is good) (genre: sci- fi/steampunk/coming of age) set in a world where humans trade their bodies for android bodies as a way to gain immortality. A young boy travels to different planets with a mysterious female companion.
Please Save My Earth - (genre: fantasy/drama/tragedy) this one is about 6 modern day teenagers who are reincarnated and had previous lives in another world.
Fushigi Yuugi / Mysterious Play - (genre: romance, fantasy, time travel) if there is one anime i would like to see in live action as an asian drama ( i don’t care if it’s a cdrama or kdrama) it would be this one. A young school girl gets trapped in a book and she travels to a fantasy world set in ancient china where she plays a key role in saving that world. She becomes the phoenix (suzaku no miko) priestess surrounded by several protectors (suzaku seishi) and they have to fight another priestess (the dragon priestess, who has a set of protectors of her own); female lead falls in love with one of her protectors.
They Were Eleven - (genre: sci-fi) this one is a murder mystery set in space. I think it’s loosely based on Agatha Christie’s And Then There Were None.
Ogre Slayer (genre: horror, drama) - this one is good if you like horror.
Most of these recs are old; i much prefer the way anime used to be than the way it is now, although i am sure there are still some good animes out there. Also, most of these (except for fushigi yuugi) have little romance. Just my opinion, but when it comes to romance, i think that the korean/chinese/thais do it better. The japanese are good with non-romance related dramas and anime.