It's kind of hard to just pick three for each category because normally you study a few films for specific subtopics, like lighting in cinematography, or particular types of storytelling, or different techniques in directing. So my top three to study are really just off the top of my head/favourites kind of.
- Amelie (Bruno Delbonnel), for colour theory, composition, and lighting
- Days of Being Wild/In the Mood for Love (Christopher Doyle), for blocking, composition, colour
- Snowpiercer (Hong Gyeong-Pyo), for composition, lighting
- MASH (Robert Altman) - I would read up on the production of this film and Altman in general
- Kundun (Martin Scorsese) - also has great cine
- La Haine (Mathieu Kassovitz) - some really good documentaries about this production in the film's extras
- The Fall (Tarsem Singh) - not critically acclaimed but the story's a good study of unique relationships you don't normally see explored
- Little Miss Sunshine (Johnathan Dayton and Valerie Faris) - classic indie story
- Eternal Sunshine of the Spotless Mind (Michel Gondry) - best example I can think of for abstract storytelling
Also, I didn't mention Bladerunner on my cine list (including the doc in the DVD's extras) because I haven't seen it in a while and my memory of the movie's hazy, but this is often cited as important viewing as a lighting tutorial by pretty much everyone. And I wanted to second whoever mentioned the Place Beyond the Pines above - it's great for both directing (Derek Cianfrance) and cine/lighting (Sean Bobbit).