I've told you a bunch about Batman in Batman v Superman: Dawn of Justice*, but not much about Superman. I don't want to give you the impression that this is Batman's movie - it's very much a sequel to Man of Steel, and it very much follows up on stuff from that movie in big ways. In fact Chris Terrio's script - at least a draft from just before production began - deals explicitly with some fan complaints about the last Superman film.
Batman v Superman opens with Batman's origin and fast forwards to Bruce Wayne coming to Metropolis during the Superman/Zod fight, but that's just setting the stage for that character. From there the script turns its attention to Superman heavily, and it examines the implications of Superman in the real world. The main action takes place a couple of years after the end of Man of Steel, and Superman has been busy in the meantime, becoming the hero we know and love. Clark Kent and Lois Lane have moved their relationship forward to the point where they live together. Clark is still working at the Daily Planet, although it's unclear if he's loving his assignments - he's tasked with covering a Metropolis/Gotham football game when he would rather be writing about Batman.
Very early in the film Lois is in the Middle East following a story when Superman shows up to help her out. This becomes something of an international incident as other nations view Superman as a tool of the United States. China, for instance, doesn't want him in their air space. It's an interesting look at how a being like Superman would be viewed on a global scale.
What are the lessons that Superman learned from Man of Steel? Some slight spoilers follow.
When Superman confronts a big bad guy and unravels his plot, he tells the villain, "I won't snap your neck, I'm going to take you to prison," an on-the-nose response to criticisms of the first movie. Of course Superman later loses his cool and threatens to break the bad guy's back, so he hasn't quite internalized everything yet.
As for being a protector of the people - when the Doomsday fight starts General Swanwick tells the military that Superman will move the battle away from populated areas because he's a hero, and that's what heroes do. Again, this is a direct response to complaints about the first film. It's also interesting to see that Swanwick, who was tracking Superman with drones at the end of Man of Steel, is such a supporter in this film. It seems that most people - except Lex Luthor and a cautious Batman, who is prepping anti-Superman weapons - believe Superman is a hero.
A caveat on this, and my previous Batman reporting - this information comes from a script that was handed in a month before principal photography began. While the broad strokes surely will make it into the film, small details like dialogue and character moments could be cut from the script or in the editing room before the movie reaches theaters.
* the actual title