Thanks to some help on Reddit and my own efforts to read the series, I think I've discovered the best reading orders, based on how many issues you want to read. All are available on Marvel Unlimited:
Bare bones (9 issues)
This reading order tells a complete story about the passage of the Superhero Registration Act and the resulting Civil War, focusing on the two sides of the conflict, led by Iron Man and Captain America.
The Road to Civil War
The major players in the Marvel Universe fear that a Civil War will happen
Civil War
The Superhero Registration Act is passed and conflict erupts throughout the Marvel Universe
Civil War Epilogue
Everyone reflects on the fallout of the Civil War
Fleshed out (22 issues)
In addition to showing the two sides of the Civil War between Iron Man and Captain America, this reading order includes Spider-Man's side of the story being caught in the middle between them. Spider-Man has a major role in the main events of the Civil War, but it is only in his own comic series where we see his motivation and struggle with doing what he does. Also, this order adds some additional information about Captain America's whereabouts, especially since he is not as prominent in the main series as his rival Iron Man.
The Road to Civil War
The major players in the Marvel Universe fear that a Civil War will happen, and Peter Parker is recruited by Tony Stark to help.
Civil War
The Superhero Registration Act is passed and conflict erupts throughout the Marvel Universe. Spider-Man is unsure of what side he's fighting for, and Captain America is on the run.
Read these issues in publication order
Civil War Epilogue
Everyone reflects on the fallout of the Civil War
"Full" Story (33 issues)
There's another side to the Civil War: that of the everyday person without superpowers. The well-regarded Front Line mini-series tells the Civil War from the perspective of two journalists who are covering it, and includes them on the scene at many side events that present a more complete world. It also has backup stories involving the remaining survivor of the incident that started the Civil War, and short reflections on the parallels between the Civil War and other historical events.
This reading order is the same as the "Fleshed Out" reading order above, but add Front Line to the list of Civil War books. Like that one, you should read each issue in that grouping in publication order.
Extras
If there are any other superheroes you like, you can read their tie in events, but they have little impact on the main event. However, they are completely optional to read. One Redditor noted that he actually got burnt out trying to be a completionist and reading everything, so it's recommended not to.
There is also What If? Civil War, which explores an alternate conclusion to the event.
