Apologies if this has been asked before, but what comics should I read to get to know Captain Marvel/Carol Danvers? Any compilations or anything like that?
Based on the logo they've chosen to use and the fact that they're calling her Captain Marvel, there's a good bet they're using Kelly Sue DeConnick's version (the current version) as a primary inspiration. That's been collected in several trades.
The 1st volume of her run was set on Earth, and it's collected in
Captain Marvel Vol. 1: In Pursuit of Flight,
Captain Marvel Vol. 2: Down and the final issues of that run are in an Avengers TPB,
Avengers: The Enemy Within. Which is basically a Carol-centric Avengers story.
However, this movie is likely to have more cosmic involvement, based on what Feige said when he announced the film, so I personally feel her more recent run is closer to what we'll be getting in the film franchise. Kelly Sue's 2nd volume is an almost entirely cosmic one, with Carol traversing the universe, occasionally working with the Guardians of the Galaxy and all that. IMO, it's just as good a jumping on point as her 1st run. And currently, that's collected in 2 trades,
Captain Marvel Vol. 1: Higher, Further, Faster, More and
Captain Marvel Vol. 2: Stay Fly. The 3rd trade of that run is due out on September 1st.
Her 1st appearance as Captain Marvel in her current incarnation was actually in a Spider-Man arc like a month before KSD's 1st run made its debut. If you're interested in Peter and Carol's relationship (in a non-romantic sense), I'd highly recommend checking it out, as it's collected in the trade
Avenging Spider-Man: The Good, The Green and the Ugly.
For her most notable original stories from decades ago, when she first made her debut as Ms. Marvel, you could check out
Essential Ms. Marvel: Vol. 1. Some folks here might recommend Brian Reed's run with the character, as that was the most notable one to date before Kelly Sue DeConnick came along, but I personally wasn't a fan of the way he wrote her, so I'll leave that to someone else, lol.
It's worth noting that after
Secret Wars is over, the character (along with the rest of the Marvel Universe) is starting a new run, this time with the Agent Carter showrunners writing her title, so ANOTHER new
Captain Marvel #1 is on its way in a couple months. I'm guessing it will give us an even better idea of their plans for the film version of the character since Kelly Sue said the reason she decided to leave was because she didn't want to commit to 3-years of lead-up (and probably studio influence, lbr) to the movie.