a lot of it was filmed in Cleveland. I'd considered trying to be an extra in the movie--some of it was filmed before I started at my current job and had all kinds of free time--but every time they needed people for the background, it was either super hot (I believe they filmed at a local beach...when it was over 90° for about a week straight) or I found out after the fact.
If I'm not mistaken, the scene in the promo where Superman and Lois are kissing in mid-air was filmed at the old arcade downtown. Not a lot of stores left there anymore, which is a shame because the building is well located and has stunning architecture. I remember a couple decades ago frequenting the food court when I had federal jury duty. Now there's a few restaurants and shops (and a post office) but most of the upper floor is part of a downtown hotel.