Wear a Watch
Don't sit there obsessively watching the second hand tick. But be aware of how much time the movie spends with its important moments, and what happens between those moments.
If you get into a habit of doing this, you'll develop an innate understanding of pacing and structure. You'll gain a better understanding of the momentum of good movies, based on how it's structured from minute to minute.
You'll notice plenty of movies that adhere to the three act structure that's taught in all the books.
But you'll also start to find movies that break those rules, and you'll start to understand how you can be creative with structure while keeping the audience interested.
Make Predictions
Sometimes you'll be right, and that's great.
More often, you'll be wrong.
Pay attention to when you're wrong, and think about why the plot progressed the way it did, and what problems your ideas may have presented to the writer (or the executives in charge of making sure the movie makes money).
When comparing and contrasting your predictions with what actually happens, think about surprise, character development, conflict and marketability.
Great movies are jam packed with surprises. They feature character development and conflict expertly interwoven to create a satisfying movie going experience for the audience.
Note: This tip works for TV too. Try it with legal procedurals and sitcoms too, and you'll start to understand a variety of mediums in exciting new ways. You can even guess the punch lines on late night monologues!
Track connections
As a screenwriter, these connections have the most to teach you. Look for thematic unity in characters, conflict, structure, setting - - everything.
This requires careful thought. You have to pay attention to every detail of the movie if you want to start seeing the connections. But at first, just to pay attention to the main character.
Track the decisions that they make throughout the course of the movie, and look for how those decisions define that character and the world/people around them, and pay attention to how it all runs together thematically.
Know when to relax
But as you're researching your next project (or if you're feeling ambitious), watch movies (and read scripts) with this information in mind. You'll come away with a better understanding of why the screenwriter did what they did, and what makes movies work.
How does watching movies help you improve as a writer? Comment below.