there's some contention about this. personally, i usually:
-
do not use any soap or detergent.
-
do not use any steel-wool nor other strong, harsh or abrasive scrubbing tools.
-
use a sponge or wash cloth.
-
wipe away all grime and food particles.
-
as needed, use a non-scratch scrubber or pad to remove stubborn bits.
-
rinse with clean water.
-
then put the cast-iron on a burner or in the oven at the lowest possible heat, just long enough to dry out most of the residual moisture.
the goal is to remove food particles but leave a thin sheen of oil/grease gently "enameled" onto the pan. this coating is called "seasoning"; it is your natural non-stick coating. a good black seasoning is built up gradually a little each time you cook with and clean the pot.
to enhance and develop this coating, try to reserve you cast iron for items fried with oil or butter, and otherwise cooking fatty foods. and to preserve it, avoid cooking highly acidic foods, like tomato sauce, as this can quickly eat away your seasoning.
perhaps the best thing about cast iron is it's essential resilience. if you're like me, you'll occasionally damage the seasoning (e.g. by leaving the drying pan on the burner too long). while this is a drag, it just means you’ll need to carefully clean out the cast iron, then begin seasoning it again. it time, it will look good as old!