It depends on what interests you. My best friend (other than my spouse) lives in Brooklyn, and used to live in Washington Heights in the very northern part of Manhattan. Both these communities are loaded with immigrant populations (my friend is from Niger, West Africa) so I found both my visits have really been through an African interface.
Very cool actually, because I've never really done any tourist things there. Except, I suppose, spending the day by myself in the Metropolitan Museum, but even there I spent more time looking at African art than anything else.
Manhattan's great for walking in. My advise is wander, and do a lot of it. I actually discovered the greatest musical instrument store ever by wandering around in Greenwich Village the first time I went. I was so lost, that the only way I found it the second time was by calling my wife and having her look it up on the internet for me.
I ended up with a 30 to 40 year old balaphone for less than $100 for the effort as well!
