Sttew
Brownlow Medallist
To do any big city justice you need to immerse yourself, stay there for more than the 3 - 4 nights, and as you say, do what the locals do.The problem with many European Cities and Big US Cities such as NYC is you can spend your limited time trapped exploring tourist traps. Over priced, full of other tourists and as such unfriendly locals who are trapped in it.
To really enjoy these countries and get the most out of them, you really need to try and get outside the tourist spots to see the real destination. For example, I've heard many times Venice is 1000s of times better when you leave the canals and go to where the locals are.
I also prefer to base myself in small to medium cities. Last year we had five days in the Netherlands before flying home and stayed in Utrecht instead of Amsterdam.