"I go shopping."
I think shopping is a location in this sentence, sort of. It's both a location and an activity. It's kind of like "let's go to where they are selling things" - but that's only my understanding, and I could be wrong.
買い = かい = kai = "buy"
物 = もの = mono = "thing"
So, I understand 買い物 to be a noun meaning "thing(s) to buy", and if that's the case, this sentence means you are going to go where the things you are going to buy are.
Whether it carries that "feeling" in Japanese when it is spoken, I do not know. When a Japanese speaker says「買い物」they need not internally model it as "things I am going to buy"; it seems likely that the word takes on the meaning of the activities they'll be doing in order to buy those things: entering stores, comparing and selecting items... in other words, "shopping". When I say "I'll drive" I don't typically picture the oxen I'm going to be whipping.
買い物 is noun. 買い物する is verb. (maybe.)
学校に行く。go to school.
食事に行く。go to meal.
both are okay following :
デパートへ買い物に行く。go shopping at the department.
デパートに買い物に行く。go shopping at the department.
'direction' is へ.
東京へ行く。I go to Tokyo.
(But we can say: 東京に行く。I go to Tokyo. )
"go to meal" - interesting. In English we do say "go to lunch" (or dinner, or breakfast) but never just "go to meal" for some reason. (But we might say "I will go eat".) Would you ever say 食べ物に行く? Maybe that sounds like "go to meal" does in English.
So, in a similar way, we do say "go shopping", but shopping is a verb. If we wanted to get closer to the literal Japanese usage, I think it would be maybe "go on a shopping trip". (But we don't need to do that, "go shopping" is simpler.)
What is 買い物 exactly? Is it the whole journey of shopping? Is it...merchandise? Is it the place that you shop at?
I guess it's not a place: as you have shown me, if it were, we could use へ. Right?
Would you be likely to use 買い物する？Would it mean about the same thing as 買い物に行く or something different?