General cleaning:
* Clean
* Rinse
* Scrub
* Sponge
* Bathe
* Launder
Specific types of washing:
* Clothes: Launder, cleanse, purify, sterilize
* Dishes: Rinse, scrub, scour, clean
* Car: Clean, detail, wax
* Hair: Shampoo, cleanse, rinse
* Body: Bathe, shower, cleanse
* Floor: Sweep, mop, scrub, vacuum
Figurative meanings:
* Remove: Eradicate, eliminate, obliterate
* Cleanse: Purge, purify, detoxify
* Disappear: Fade, vanish, dissolve
More informal words:
* Clean up: For general cleaning
* Give it a good wash: Emphasis on thoroughness
The best synonym will depend on the specific situation and the desired level of formality.