Being a New Yorker by choice I feel strongly about my city and want you to get the most out of your stay here; so I feel the need to comment on some of the suggestions:
- Times Square --> only tourists go there, nothing to do or see there unless you like paying through your nose, big shiny advertisements and large screens or are on your way to a Broadway show;
- Chinatown --> the quintessential immigrant neighborhood; Canal Street is NOT the real Chinatown, that's where the tourists go; orient yourself more towards the North and East, like Grand St, Delancey St, and perpendicular to that Mott St. and Elizabeth St.; you can make great discoveries in these streets and feel like you've left the US!
- Central Park --> as a New Yorker it's what keeps us sane, as a visitor you might not have the same emotional connection to it, but there are many beautiful spots to check out, especially in the North, where most tourists don't go; maybe you can rent bikes or negotiate a good rate with a pedicab rider who should be able to take you all around the park in 1-2 hours.
- Empire State Building --> has still the best views of the city since it's the highest building and right in the center of downtown; just go on a weekday right after opening or an hour or two before closing. I've been there 3 times and will go back anytime I have visitors, you can't beat the all-encompassing view; Top of the Rock is great too, but you don't see as much of downtown and the surrounding area.
- Statue of Liberty --> It's turned into a tourist trap since you can't go up the statue anymore; if you want to see it from upclose without wasting too much time though take the Water Taxi instead of the Staten Island Ferry. The SI ferry is free but will take you at least 1.5 hrs round-trip, whereas the Water Taxi Statue of Liberty downtown express is $10 each but takes you straight down from 44th St. to the statue for a photo op and then lets you off at Battery Park.
My point is simply that you can and should enjoy all the famous sites, just do it like a New Yorker would ;-)
Finally,
New York is still the safest big city in the US. I feel very safe in Manhattan, no matter what neighborhood or time of the day/night; the subways are also very safe. It's nothing like in the 70s/80s. This is a very different city now from what it was then I hear from New Yorkers (by some with remorse).
You'll have a great time here!