Over here we use ‘washing-up liquid’ for hand-washing dishes. In fact, after a meal we ‘do the washing-up’. (Or load the dish-washer.)

In my household, (childhood one and current) we did "the dishes" (or loaded the dishwasher)
as children, we set the table, before dinner, (starting with a cloth for the table) after dinner, we cleared the table and did the dishes. Part of clearing the table included careful removal of the table cloth, which was furled out a window to remove the crumbs, before being folded for reuse, or tossed in the hamper for later washing.

thankfull, my mother read a pamplet from NYC board of health, that said it more sanitary to let dishes air dry, than to wipe them, (dry them with a cloth), so the sat in the drain board, and latter, my mother would empty it. (the cloth in question-- in our kitchen was always a tea towel- but nowdays, i am as likely to say kitchen towel.
paper towel were a luxury, and used scantily, nowdays, for ecological reasons, i too, use very few.. a roll of paper towels last me about 3 months!

to pick up hot pots, we used a potholder..(i know there are other term... lets hear 'em!)