The last few years of Captain America can be split up into Sam Wilson, Secret Empire, and the current series. For new readers, I would probably jumping into the current series, Captain America (2017). It's a fresh but classic take on the Steve Rogers you are probably expecting.
However, if you're up for something different, in 2014 Sam Wilson, the Falcon, was given the mantle of Captain America to carry forward, and over the next few years, with Steve Rogers, a really incredible story unfolded which was concluded in the big summer event comic of 2017, Secret Invasion. This is definitely one of my favorite stories in comics, but it's not your standard Captain America story.
The beginning of that epic is laid out next, followed by the current Captain America series which is a great starting point and definitely a much more familiar experience, and then a few short stories for those looking for something more self-contained.
Steve Rogers has been aged, 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.
All-New Captain America (2014) paperback reading order:
This is where the backstory for the the 2017 Marvel event really starts to be told. Two books, Steve Rogers and Sam Wilson, tell parallel stories. You can easily read just one, or both side by side. Captain America Steve Rogers / Sam Wilson (2016) paperback reading order:
Secret Empire (2017)
A true return to form, the current 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.
Captain America (2017) in paperback reading order:
If you're interested in single book that tells a completely self-contained and story, here are a few good ones from the last few years: