Dittos on the STL-style pizza. Some like it, some don't. But, it's definitely unique, and there is no better example than Imo's Pizza, which are all over the place.
As for the gun-friendly part ... yes and no, depends on where you go. You most definitely cannot carry on public transportation (Metrolink trains and buses). In fact, you can't even carry onto their platforms. If you venture over here to the Illinois side (lots of cool things here, too, some mentioned above), out-of-state permit holders cannot carry outside of your vehicle.
Most importantly, stay the hell out of North St. Louis City (and a fair portion of the nearby County). It's a shooting gallery up there, and has gotten much worse since the Ferguson fiasco.
Be safe, be smart, and enjoy your visit.