Here's a list of a few notable anime directors and the work I consider worth watching. Certain directors has a very distinctive style while others are more subtle. If your favorite anime isn't on here, it's because I didn't like it.
Categories are
Favorite
My personal favorites. A must watch.
Worth watching
Not a must watch but still above average.
Consider
If you've seen everything else and is still wanting to see more.
*Every anime that they had a major role in but did not direct themselves, I'm putting into consider. Consider these bonus entries*
Have not seen
Satoshi Kon
Notable for his editing style that blends reality and dreams together.
Any film of his is a good starting point since they all delve into similar topics ....
Favorite
Millennium Actress
Perfect Blue
Paprika
Tokyo Godfathers
Consider
Paranoia Agent (TV)
Memories (non-director role)
Yamada, Naoko
Ever since Tamako Love Story she has made very intimate and quiet films. I think she's the stand out director in Kyoto Animation.
Has progressed immensely and truly found her own void after Koe no Katachi....
Favorite
Liz to Aoi Tori
Koe no Katachi
Worth watching
Tamako Love Story
Consider
Clannad (TV)(non-director role)
Clannad: After Story (TV)(non-director role)
The Disappearance of Haruhi Suzumiya (non-director role)
Have not seen
K-on (TV)
Miyazaki, Hayao Studio Ghibli
Beautiful backgrounds. Detailed animation. Miyazaki has said that he does not write a storyboard for his movies but instead he focuses on creating captivating scenes and only after creating enough of them does he see where the film is heading.
Favorite
Spirited Away
Princess Mononoke
Ponyo
Nausicaä of the Valley of the Wind
Worth watching
Lupin III: Cagliostro no Shiro
Porco Rosso
Consider
Kiki's Delivery Service
The Wind Rises
Castle in the Sky (Laputa)
My Neighbor Totoro
Howl's Moving Castle
Have not seen
Future Boy Conan
Takahata, Isao Studio Ghibli
Minimalism.
Very interesting comparing Takahata to Miyazaki since they're so different....
Favorite
Grave of the Fireflies
The Tale of the Princess Kaguya
Only Yesterday
Worth watching
My Neighbors the Yamadas
Pom Poko
Consider
Jarinko Chie
Gauche the Cellist
Have not seen
The Little Norse Prince
Akage no Anne
Rest of Studio Ghibli filmography
Favorite
The Secret World of Arrietty (Yonebayashi, Hiromasa)
The Cat Returns (Morita, Hiroyuki)
Worth watching
Whisper of The Heart (Kondou, Yoshifumi)
From Up Poppy Hill (Miyazaki, Gorou)
The Ocean Waves (Mochizuki, Tomomi)
Hosoda, Mamoru
Favorite
Wolf Children
Worth watching
Summer Wars
The Girl Who Leapt Through Time
Consider
The Boy and the Beast
One Piece: Baron Omatsuri and the Secret Island (Movie 6)
Have not seen
Mirai no Mirai
Yuasa, Masaaki
Overly expressive and creative animation.
Favorite
Mind Game
Ping Pong the Animation (TV)
The Tatami Galaxy (TV)
Worth watching
Devilman: Crybaby (TV)
Kaiba (TV)
Kemonozume (TV)
Night Is Short, Walk on Girl
Consider
Kick-Heart
Have not seen
Lu Over the Wall
Kimi to, Nami ni Noretara
Shinkai, Makoto
Very beautiful backgrounds.
Favorite
5 Centimeters Per Second
Worth watching
Your Name.
The Garden of Words
Consider
She and Her Cat: Their Standing Points
Have not seen
Weathering With You
Shinbou, Akiyuki
Emphasis on colors and various animation quirks.
Favorite
Monogatari Series (TV)
Kizumonogatari
Mahou Shoujo Madoka★Magica (TV)
Worth watching
Sayonara Zetsubou Sensei (TV)
Consider
Magical Girl Lyrical Nanoha (TV)
Have not seen
Soredemo Machi wa Mawatteiru (TV)
Paniponi Dash! (TV)
Arakawa Under the Bridge (TV)
3-gatsu no Lion (TV)
Hidamari Sketch (TV)
Watanabe, Shinichiro
Favorite
Cowboy Bebop (TV)
Kids on the Slope (TV)
Samurai Champloo (TV)
Space☆Dandy (TV)
Consider
Zankyou no Terror (TV)
Have not seen
Macross Plus (TV)
Katsuhiro Otomo
Mangaka behind famous manga such as Akira and Domu.
Favorite
Memories
Akira
Consider
Steamboy
Hideaki Anno
Favorite
Neon Genesis Evangelion (TV)
Neon Genesis Evangelion: The End of Evangelion
Worth watching
Evangelion Rebuild Movies
Re: Cutie Honey (OVA)
Gunbuster (OVA)
Consider
His and Her Circumstances (TV)
Have not seen
Nadia: Secret of Blue Water (TV)
Oshii, Mamoru
Favorite
Mobile Police Patlabor 2: The Movie
Angel's Egg
Ghost in the Shell 2: Innocence
Ghost in the Shell
Worth watching
Patlabor: The Movie
Consider
Jin-Rou (non-director role)
Urusei Yatsura Movie 2: Beautiful Dreamer
Dezaki, Osamu
Have not seen
The Rose of Versailles
Aim for the Ace!
Tomorrow's Joe
Black Jack
Takarajima
Space Adventure Cobra
Ikuhara, Kunihiko
Have not seen
Mawaru Penguindrum (TV)
Revolutionary Girl Utena (TV)