Have you been shopping or gone shopping?

It all depends on whether you have come back or not!

Let's look at this example:

Yes.

“He’s been shopping” tells us that he’s gone and come back again.

“He’s gone shopping” tells us that he’s still there.

Confusingly they can be used together in the same sentence with the phrase "been and gone". An example would be:

Q: "Has the postman been yet?"

A: "Yes, he's been and gone"

Meaning that the postman has made a delivery and left again.