As a native New Yorker, I think that any place a tourist wished to go to get a sense of the city would be fine. It is possible that to wander into East New York, or parts of the South Bronx, or Bushwick or even Brighton beach (among others) might be tempting fate to some degree, but during the day its pretty safe. But then again, 99% of tourists have little interest in such places in the city. Within Manhattan, during the day, one can go anywhere. 125th Street and the Apollo theatre (not to mention Slick Willie's offices) are something that more tourists should see, as are the lower east side.
You will have a great time in NY.
It's a fantastic city and I think very different from the rest of the States. Very vibrant, full of life and diversity. The best way to get to know the city is to just stroll around and stop at cafes, restaurants and bars for long or short breaks.
In my view NY is very safe, walking back to the hotel in the middle of night was no problem at all, rather quite romantic as I remembered it
If you want to be able to walk to the most stuff, I would say Union Sq. area is ideal becuase you can walk all over midtown, soho, chinatown, east village, west village - all within about 30 minutes.
I totally agree that Union Square is one of the best areas to stay to be able to get around easily around the city. Half the time I go I stay with my aunt, who leaves two blocks from there, old roomates or when I want to do my thing I stay here...extremely reasonable for NYC and clean, which is very important.
By the way, Union Square is located between 14 and 17th Street and 4th and 5th Avenue.
You should very much consider venturing into brooklyn. Not only is the drive over either the brooklyn bride or the manhattan bridge one of my favorite things to do (so beautiful), brooklyn is an amazing place. if you travel into park slope you have the brooklyn museum of art. Right off the brooklyn bridge on flathbush you have Brooklyn Arts and Music [url=http://www.bam.org. which is a beautiful building that has plays, movies, music, and dance performances. But there is also the promenade in brooklyn heights which gives you an amazing view of the downtown area of brooklyn and the grimaldi's pizza is right there too. I have seen some of those city tour buses around that area, so if you are unsure about trekking into brooklyn on your own maybe you should check out one of those.