Something that I am increasingly finding more and more credence for is the idea that animated films, primarily well-regarded 2D animated features, can help with shot composition more easily than live-action can: at least as an additional guide. Because I think some directors will move into the field without a formal understanding of shot composition and an understanding of the meaning behind certain shots. And this can result in a very cookie-cutter and average looking film rather than one that is brimming with an artistic voice and specific compositional and editorial choices. So studying 2D animation can be quite useful because unlike live-action, animation is based on artwork, and artwork, like paintings or drawings, are often approached on a deep compositional level because every single element in the frame must be drawn by hand and must have a purpose. And because 2D is flat and lives on a finite level, you are locked into a single angle for each new shot that you make since the backgrounds cannot be turned. In this way, you end up making more defined choices rather than changing your mind and going about your choice of angles more nebulously.
So I would highly suggest to study up on most of the Studio Ghibli films: Nausicaa, Laputa, Kiki's Delivery Service, Porco Rosso, Princess Mononoke, and Spirited Away. And then go and study something like Cowboy Bebop for something a bit more geared towards American Live-action sensibilities.
But for live-action, I think it's a safer bet to not look towards the greatest of masters at first, as their work is extremely difficult to achieve. So there's no "practical" purpose in studying their work first, since you can't easily go right out and start practicing upon it.
Instead, I would look at films by people like J.J. Abrams, Christopher Nolan, Chris Columbus, David Lynch, Guillermo Del Toro, and of course Steven Spielberg, mainly because these directors are both mainstream, but also have a somewhat more achievable style that one can study from and try to emulate while practicing the craft: as long as you have access to the right lens lengths of course. But even then you really just need a mid-prime (50mm), a short telephoto (14-42 or so), a wide adapter for the short prime, and a longer telephoto (about 45-85 or there-abouts).
I was also very impressed with feature film new-comer Damien Chazelle and his film "Whiplash:" which was only made for a measly $3,000,000. So definitely give that a watch as well.
For things to look for, I would try to study things scene by scene to see how they are structured editorially--timing, eye-lines, overlapping dialogue--and then break that down into the smaller elements like shot composition, the lighting and how it helps your eye to focus on certain things in the frame or how it affects the mood of the scene, and then study the sound effects and music and how they work to emphasize or improve on what you are seeing. If you're also concerned with performances, you can look at that perhaps in the middle, or last, depending on your interest areas.