First off, in the US a bathroom is a place with a toilet and sink. The actual tub/shower is optional. (When selling homes the toilet-only room is called a "half bath" while the one with tub is a "full bath", and it's a "three-quarters bath" if it's got toilet and shower but no tub.) However, in businesses and other public buildings the toilet room is more commonly referred to as a restroom.
And even when the "facilities" are no more elaborate than a shed with pit underneath it, the typical American English speaker may very well refer to it as "the bathroom" out of habit, though in this case the term "toilet" is more apt to be used.
When describing his intent to bathe his entire body, the typical American English speaker would say he was going to "take a bath". If he intended to shower only, he might instead express that qualification by saying he was going to "take a shower". It's vaguely possible he might express an intent to "bathe", but that's a rather quaint way to say it.
If the typical American English speaker needed to perform an excretory function he would say he needed to "go to the bathroom" (possibly substituting some euphemism for "bathroom"), or, particularly before/after a meal, he might say he needed to "wash up". And it would not be unusual for a host to courteously ask if a guest would like to "wash up" before a meal (or perhaps simply after arriving in the house following a lengthy journey). On re-reading this it occurs to me that "use the bathroom" would be more idiomatic (and slightly more polite) than "go to the bathroom", since "go" implies excretion while "use" is more ambiguous.
(But note that these are "typical American English" terms, and it's entirely possible that different terms are used in West Boston or some such, as is common with domestic terms.)