"I am going to a bathhouse to bathe."
Translation:목욕하러 목욕탕에 갈 예정이에요.
예정이다 actually means expected to, due to, scheduled to. So it has to do with a set (confirmed) plan. It tends to be used in a more business context.
계획이다 is used for planning to do something (but nothing confirmed).
But it seems colloquially the two expressions are now interchangeable.
Going to + verb only indicates the future if used with an appropriate time adverbs indicating as such: later; in the afternoon; tomorrow etc.
e.g. I am going to the bathhouse later = I will go to the bathhouse later.
Without the adverbials, the present progressive will indicate an action currently in progress.
e.g. I am going to bathhouse = I am on the way to the bathhouse
To stress that it is a near future action in this case, the full expression "going to go to" has to be used.
Personally I think "be going to" is fine as it expresses the wish to keep to a future plan decided before the moment of speaking
"will" expresses a future plan made at the very moment of speech which may or may not be subjected to changes. Open promise.
The difference between the 2 expressions is not so significant and may be used interchangeably at speaker's preference.
The closest translation to this example is: I plan on going to the bathhouse to bathe