Also, walk. Walk in random directions, without a plan. See what's really going on!
In my opinion, there's no other way of truly experiencing another culture. Taking a picture in tourist-packed place is surely not the way to experience anything.
I realize that's difficult to spend a month in another country due to other obligations (kids, work, etc.) - not sure how are we going to solve that, ever (or even if it needs "solving" at all). That's just how society is structured, for now.