мыться - lit. 'wash[one]self' - bath or shower.
умываться - your face. Also that's what cats do.
умывать - someone else's face. Also that's what cats do with kittens and each other.
these two can be applied to other things, actually, including metaphorically.
мыть голову (hair)
мыть пол (to wash floor, with water).
подметать - to sweep (floor or street) with a broom.
мести, заметать etc - related verbs.
убирать(ся), делать уборку, and synonymous "приводить в порядок" (put in order) - mostly about spaces, like rooms. Mostly about overall disorder, or large litter, but may include dirt too.
убирать also means 'remove'.
протирать, вытирать, вытираться - lit. 'rub through', 'rub out'
1) cleaning surfaces - dusty mirrors or bookshelves, stained table - with cloth, wet or dry, in your hand.
2) drying anything (dishes, hands, body, even floor) with a towel or a piece of cloth... as long as it involves Rubbing movement.
3) if you spilt coffee on the floor, you "вытираешь пол тряпкой",
стирать - washing clothes.
чистить - literally "clean".
1) removing dirt when it needs considerable phisical effort.
2) in number of situations when other verbs are misleading
- metal surfaces, that you want to make shining. Or when stains are hard to remove.
- your coat with a brush.
- birds do this with their feathers.
- dry cleaning.