I recommend starting Captain America comics either with the fantastic current Cap storyline by Nick Spender, or a self-contained book with classic Steve Rogers.
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, Captain America: Sam Wilson, and Captain America: Steve Rogers in paperback reading order:
However, like Nick Fury said in the Avengers movie, 'people might just need a little old fashioned'. Below are three totally independent stories with very classic Steve Rogers Captain America stories.