So, the translation 'I am making/(taking) a tour' should also be accepted, shouldn't it?
I don't think so (note that I am not a native speaker). I suspect the point of this sentence is to introduce us to ideomatic expressions using "tage" that are not adequately translated using "take". So, likewise "jeg tager afsted" means "I am leaving."
Why not, I am on a trip?
Why not: 'I'm going for a trip'?
Hvorfor kan man sige ikke "I go on a tour" ? Må man sige "trip" og ikke "tour"?