You wouldn't necessarily say "do shopping", but you can certainly say "let's do some shopping".

Yes, you can 'do some shopping'.

Thanks!

"Do shopping" sounds awkward. A more correct way of saying it would be "shop". For example, instead of saying "Let's do shopping" you'd say "let's shop". The same thing would apply to "reading" or "cleaning". "Let's clean/read" sounds more natural than "Let's do cleaning/reading". Throwing in "some" to get "let's do some shopping/cleaning/reading" sounds natural as well.