I really want to read about this character before the movie is released but have no idea about her and where to start. I do know that the first Captain Marvel was a man named Mar-Vell and that Carol was once known as Ms. Marvel but don't know much about her origin story. Which comic series tells us about her?
The Marvel Cinematic Universe does not correspond exactly to the comic book universe. In addition, there are so many incarnations of Carol Danvers over the last fifty years that it would be more confusing than helpful to study her history. If it were me, I'd go see the movie without taking all the baggage along for the ride. If the other movies are any guide, the finished product won't follow the original stories. Each movie is intended to stand alone.
To answer the question, if you want a sense of what the latest incarnation of Carol Danvers is like, I'd read Captain Marvel, volume 7 (2012-2014), and volume 8 (2014-2015). You'll learn as much as you need to know about her origin.
As noted in the link Valorum provided in the comments, Carol Danvers' origin story isn't readily defined.
She first appeared as a supporting character of Marvel's Captain Marvel (MARVEL SUPER-HEROES 12-13, CAPTAIN MARVEL 1-18) in the 1960's. Technically, she gained her original power set in CAPTAIN MARVEL 17; however, at the time, no one knew that.
We only actually found out the she had gained powers back then in her first solo series, MS MARVEL, from the 1970's. Her original powers included flight, super-strength, and some sort of psychic ability that gave her visions of the future (referred to as her "seventh sense").
After a 23 issue run of the solo title, and a role as first a guest-star, then a member, of the Avengers, she was written off in a really bizarre storyline running through AVENGERS 194-200 (or so). About a year later, in AVENGERS ANNUAL 10, her powers were stolen by Rogue, and she lambasted the Avengers, for the events of that story.
From there, she guest-starred in UNCANNY X-MEN for a while (from around 157 through 166 or so). In those stories, we find her second origin. The one thing that Rogue did not taken from her was a general enhancement of her genetic structure, leaving her in part genetically a member of the Kree (the alien race Marvel's first Captain Marvel hailed from). In an encounter with the alien race called the Brood, experimentation granted her a new power set - the ability to absorb, store, and shoot off energy. This also gave her flight (and probably super-strength), as well as the ability to survive in space without a space suit (apparently, at least). She changed her codename from Ms Marvel to Binary, and spent the rest of the 80's and most of the 90's as a member of the Starjammers, in adventures that (for the most part) happened "off-screen" - she appeared as a guest-star in the X-Men family of books, and on very rare occasions, other books as well.
Side note: During this time, two others took up the name "Captain Marvel". First, Monica Rambeau, who gained the ability to transform herself into any form of electromagnetic radiation, and joined the Avengers. Then, Genis-Vell, posthumous son of Mar-Vell (the Kree who used the name first in the Marvel universe).
In the late 90's, Carol returned to Earth, rejoined the Avengers, but found her powers had weakened greatly. She could still absorb energy and spit it back out, but most of the time she was back to her "Ms Marvel" abilities, flight and super-strength (no psychic "seventh sense" this time). She changed her nom de guerre again, from Binary to Warbird. This was seen mostly in the first year or so of AVENGERS volume 3 (issues 1-12 or so, plus a couple of tie-ins) circa 1998-9 (by Kurt Busiek and George Perez; personally, my all-time favorite run of the that title).
She went back to using "Ms. Marvel" just before the first Marvel "Civil War" storyline. And, a few years later, decided to upgrade to "Captain Marvel" (with a bit of discussion in her first CAPTAIN MARVEL #1 issue, circa around 2012 or thereabouts).