Old English bæð "an immersing of the body in water, mud, etc.," also "a quantity of water, etc., for bathing," from Proto-Germanic *badan (source also of Old Frisian beth, Old Saxon bath, Old Norse bað, Middle Dutch bat, German Bad), from PIE root *bhē- "to warm" + *-thuz, Germanic suffix indicating "act, process, condition" (as in birth, death). The etymological sense is of heating, not immersing.
The city in Somerset, England (Old English Baðun) was so called from its hot springs. Bath salts is attested from 1875 (Dr. Julius Braun, "On the Curative Effects of Baths and Waters"). Bath-house is from 1705; bath-towel is from 1958.
also bath-room, 1780, from bath + room (n.). Originally a room with apparatus for bathing (the only definition in "Century Dictionary," 1902); it came to be used 20c. in U.S. as a euphemism for a lavatory and often is noted as a word that confuses British travelers. To go to the bathroom, euphemism for "relieve oneself; urinate, defecate," is from 1920 (in a book for children), but typically is used without regard for whether an actual bathroom is involved.
"soaked or softened in water," 1820, earlier "resembling something that has been boiled a long time" (1590s), originally "boiled" (c. 1300), from Old English soden "boiled," strong past participle of seoþan "to cook, boil" (see seethe). For sense evolution from "heat in water" to "immerse in water" compare bath.