"Let's go shopping today."
What is the function of "shopping" in the above sentence ?
Is it an adverb ?
It seems to me that there should be a "for" before shopping in the sentence .
To me, "shopping" seems like a noun. I am happy to be corrected on this though!

"Let's go for shopping" isn't right. (However "Let's go for a walk/beer/coffee/etc." is OK. In theory you could say "Let's go for a shop" but in practice this would be unusual.)
There seem to be two ways to analyze this kind of sentence:

(1) Let us go shopping.

(a) "shopping" is a participle that modifies "us."

(b) "shopping" is a gerund that is the object of a preposition that used to used many years ago:

on + shopping > a-shopping

(i) The prepositional phrase (on shopping/ a-shopping) modifies the verb "go"