But first, not that Civil War: The American Civil War of the 1800s is not related to this story, except that the heroes of this story are turned against one another, in internal conflict.
Okay, so what is a comic event? An 'event' is like the comic equivalent of the Avengers movie. It involves a lot of characters that have their own stories. You can watch it by itself, but there are also a TON of things you can watch which give it context, like other movies or even TV shows. But you don't need to watch them all, just like you don't need to read every issue in a comic event.
How do I read them? The easiest and cheapest way is Marvel Unlimited, a service like Netflix for comics, where you can read anything over 6 months old for $10 a month. This great list has every issue on the service. However, Marvel Unlimited does have problems, and some people just want to hold some beautifully printed books in their hand, lend them to friends, or get frustrated with the Marvel Unlimited web reader. In that case, I'll be linking to printed copies of all the issues below.
Here are the biggest parts of the Civil War event:
These seven issues, Civil War #1-7, are the core of the event. If you just want to read a single story, this is what you need. You can read it by itself and it will make sense. This is the 'Avengers movie' of the whole event, a self-contained story which everything else below here ties into. These issues were written by Mark Millar, a writer well known for many series including The Secret Service, Wanted, and Kick-Ass which became the films Kingsman: The Secret Service, Wanted, and Kick-Ass.
I highly recommend also reading the six issues included in Road to Civil War, which lead into the main event. This collection gives backstory and context, and begins with my personal favorite issue of the entire event, New Avengers: Illuminati. It also includes more backstory on the Peter Parker angle of Civil War and his relationship with Tony Stark, so it looks like it will be a great story to read before the movie.
Civil War: Captain America/Iron Man takes place at the same time as the seven issues of Civil War. As the two titans of the story, they are considered the two most important tie-ins. Personally, as a massive Spidey fan, I like to add The Amazing Spider-Man: Civil War. Unlike Road To, these issues are read mixed-in with the main issues of the event, but can also be read stand-alone if you want. Think of them like the first three Iron Man movies and the first two Avengers movies. They are meant to be seen in order, but they also work as their own story.
Civil War: Front Line tells the story of reporters Sally Floyd and Ben Ulrich. You might remember Ulrich from the Daredevil Netflix series. With the main Civil War storyline very short and full of action, many people felt the series lacked depth, and didn't always make sense. Frontline is hailed for providing character depth, cleaning up loose ends, and filling in the holes of the main Civil War stories, as well as being very well written.
Wolverine: Civil War — While everyone else fights amongst themselves, Logan decides to go kill the people responsible. Because he's Wolverine.
Black Panther: Civil War — Exists at the perimeter of the Civil War looking in. T'Challa and his new wife on a diplomatic tour as the events of Civil War unfold.
Fantastic Four: Civil War — Reed Richards plays an important role in Civil War, but this tie-in is really more for hardcore Fantastic Four fans.
Civil War: Heroes for Hire/Thunderbolts — Two mercenary groups hunting down fugitive heroes.
Civil War: Punisher War Journal - Expands on his great role in the main storyline.
Civil War: Young Avengers & Runaways — More about the two teams during that time.