Cleaning cast iron should be an easy process, and is shouldn’t take more than some warm water, a bit of gentle dish soap, and a sponge or gentle (non-metal) brush to wipe out the remnants of whatever was last cooked in the pan. If your skillet is properly seasoned, and you’re using an adequate amount of fat during cooking, food should not stick to the skillet.
Remember, cast iron becomes nonstick from baked-in fat, so too much scrubbing and grease-cutting dish soap can render them dull and, well, non-nonstick! It sounds sort of counter-intuitive, but you don’t want the pan to get too clean, because it will lose its stickiness. Just clean it normally with a sponge and mild dish soap.
But things happen and food will stick from time to time. Unlike stainless steel, which you’d soak, you will need to scrub stuck food off of your cast iron skillet. DO NOT SOAK YOUR SKILLET!
To remove stuck-on food, first try hot water, mild dish soap, and a non-scratch sponge. If that doesn’t loosen all of the food, sprinkle a layer of table or kosher salt over the cooking surface of the pan, as a gentle abrasive, and rub with the sponge. If you’re still unable to get the food off, try balling up a piece of tinfoil (aluminum foil) and use that to scrub, in place of the sponge.
Once you’ve gotten the food off of the cooking surface of the pan, make sure to follow the process for cleaning cast iron, described above, to re-establish the non stick surface of the pan.