I'm a professional welder, and in order to answer your question, I'll have to pose one of my own. By Flux Core, do you mean actual flux core (core shield), or do you mean duo shield. (solid wire with a C02 gas shield)? Alot of people call Duo Shield flux core. I personally try to never use core shield welding on anything, as it is a dirty welding process. Cleanup and prepping for painting tends to be alot of work, and if what you are welding happens to be structural at all, I wouldn't recommend core shield at all, regardless of the base material. It tends to develop alot of porosity in the welds (pinholes that drastically reduce the structural integrity of the weld.)

As far as stick goes, 7018 is a good all around rod for steel. For a 3/16 " butt weld, I would recommend using a 3/32" diameter rod. 7018 is an "all position" rod. Meaning that you can weld overhead, vertical up, flat, or horizontal with it. Make sure that when you do go down and buy your rod, you bake it in your oven first at 450 degrees for about a half hour. This removes any moisture from the flux that might create porosity. And keep it in an air tight container when you aren't using it, or else you'll have to bake it every time you use it.

Hope this helps.

agree with the guy above but if its not structural then .035 flux core wire will do just fine. bevel the joints and weld away. Might have to deal with some splatter but they make a product to spray on the keep it from sticking. Or you can just knock it off. I personally wouldnt weld 3/16 with a gas shielded welder unless it a bog welder, but being a hobbiest I assume you have a lunch box machine. .035 and be done with it. I hate stick LOL