I just got back from NYC on Sunday...I had one day to see the sights. I was staying in Jamaica Queens, took the subway to Manhattan (that was a fun expereince!) to the world trade centers sight. I bought the day pass for 7$, unlimited rides on Subway, which takes you everywhere. Then I hit up MSG (Madison Square) looked around that area, Central Park south side, Empire State building, Grand Central Station, Times Square for dinner and some quick shopping. Wear comfy shoes and you'll have a great time. I've never felt more safe in a city as I did in NYC, a cop on every corner, Soldiers in some spots. You'd have to be really careless and lack street scents to get in trouble out there, I was still walking around after 11pm, taking the subway back. NYC is a world in itself, I've never seen a city so spectacular! Trust me, you'll have a million and one things to do out there and not enough time!
Now if you want a city that you won't feel safe in, come to Vancouver BC Canada, you'll be pestered by street bums, approached and harrassed by addicts, and mugged by drunk indians :| I kid you not.