If you want to get into Fate, I highly recommend reading the Fate/Stay Night visual novel before watching Fate/Zero as Fate/Zero casually spoils several very important plot twists in Fate/Stay Night and its climactic route.
Mononoke (not Mononoke-hime) is honestly the best horror I've ever seen animated since it actually keeps an unnerving atmosphere instead of just throwing gore everywhere. The visuals are also breathtaking.
Amagami SS is a romance slice of life series based off of the galge Amagami. It follows an omnibus format, where each girl gets a 4 episode arc instead of giving you 26 episodes of harem antics.
Mushishi focuses on people who are affected by enigmatic creatures named mushi and a man who travels around trying to help such people. Bittersweet endings abound and relaxing music/scenery make this a memorable view for anyone interested.
Kaiji and Akagi are both madhouse adaptations of gambling manga written by Noboyuki Fukumoto. Akagi focuses on mahjong while Kaiji focuses on made-up games like "restricted rock paper scissors" and "E Card". Both are worth a watch (even if you don't quite understand mahjong).