Captain America originally debuted in Captain America Comics #1 (1940) by "Timely Comics", Marvel's predecessor. The story of Captain America in the comics and movies mirrors the story of the character's publication history in a way I absolutely love: Captain America was originally a comic series during World War II, essentially war propaganda. In the story of Captain America, The Avengers #4 reveals he was frozen at the end of World War II until the 1960s. Outside the comics, in the real world of Timely and then Marvel comics, the character of Captain America loses popularity and is canceled in 1950. The character was reintroduced in 1964, as Jack Kirby who worked on the original series brought the character back and created Steve's on-ice backstory.
For new readers, I would probably recommend jumping into something more recent. The 2014-2017 storyline is one of my favorites in all of comics, covering Sam Wilson (the Falcon) taking over the mantle of Captain America, while a parallel story with Steve Rogers is told, concluding in one of the best finales marvel has produced.
I've also highlighted the more recent series for those seeking a more classic introduction to Steve Rogers,= and some self-contained paperbacks for people looking for quintessential Cap stories with less commitment.
If you're interested in a single book that tells a completely self-contained and story, here are a few good ones from the last few years:
Steve Rogers has been transformed into an old man and has passed on the mantle of Captain America to his loyal friend: Sam Wilson, The Falcon ( played by Anthony Mackie in the movies ). This story, with Sam Wilson and Steve Rogers, will lead into Marvel's next big event, Secret Empire, so this is a great place to jump into Marvel comics. Feel free to skip the first book, since all the rest are written by a different writer, so the second book is a great starting point as well.
As the backstory for the final conclusion, Secret Empire, begins the story is told in parallel from two perspectives: Steve Rogers and Sam Wilson. You can easily read just one, or both side by sides.
All-New Captain America (2014) paperback reading order:
A true return to form, the following run of Captain America is the one readers are probably expecting from comics. This is a great place to start reading, both classic and modern, and by an incredible team.
With only one volume out in paperback, there's not much that I can say about the current ongoing run of Captain America yet except that it is written by Ta-Nehisi Coates. Coates is famous for his writings on race relations in the United States and so after his definitive run on Black Panther, Captain America is the logical next step. Readers can expect a heavy series but one with a lot of impact.