So when you say "do some sightseeing" you put emphasis on the "doing" and it's still pretty much correct?Since there is a verb "sightsee", there is no reason to "do sightseeing". On the other hand, "go sightseeing" has a different meaning, with the emphasis on the "going".
I think we use it more than, say "defenestrate."GWB, although the verb "sightsee" exists, I don't think we use it much, do we? (I don't think I do, anyway)
So, would this work here?It is natural to use "shop" by itself as a verb.
It is natural to say "go shopping".
It is not natural, however, to say "do shopping" without some other word coming between the "do" and the "shoping":
I do my shopping at Sainsbury's.
We will do some shopping tomorrow.
This is also true of "do sightseeing", which by itself sounds odd, but as "do some sightseeing" is not so peculiar. I wonder if this is true of verbs generally...