MCU Steve is pretty darn close to 616 Steve, and this is really evident in CA: TWS. A common misconception about Steve is that he’s a government lackey used to impose the will of the US on other countries. This couldn’t be farther from the truth. Steve is loyal to the idea of America, where oppressed people can come and make something of themselves with opportunities that wouldn’t have been given to them elsewhere.
The parallel can be drawn in the scene where Fury brings Steve to the Project Insight bay and explains the program to him. Steve’s obvious disapproval is a perfect adaptation of his 616 personality brought to the big screen.
As for which comics to read, I highly recommend you read the Captain America: Man Out of Time miniseries. It is by far the best representation of Cap’s character, along with having fantastic art and a very enjoyable story (not to mention a lot of derp faces from Steve). I also recommend reading vol. 5 of Captain America. It’s where I started, and it’s where Ed Brubaker retconned Bucky’s death and began the Winter Soldier story. The Captain America and… series is also a good one to read, and it explores Cap’s relationships with Bucky (Captain America and Bucky #624 gives the history of the Winter Soldier through Bucky’s eyes),
Hawkguy Hawkeye, Iron Man, and Black Widow. If anyone else has any other suggestions, feel free to add them on!