Stay in a hostel or a B&B instead of a chain hotel. Ask the people who work there where
they like to hang out. Walk around your temporary 'neighborhood.' You will probably find your favorite pizza-by-the-slice place, a friendly bagel vendor, a deli with hundreds of menu items and a -per-plate ethnic restaurant -- all on one block. One highlight is Clinton (formerly Hell's Kitchen). This area has been cleaned up, and the best local fine dining restaurants are at Restaurant Row.
Quick Tips:
Don't be afraid to try something new to eat, like 'Egg on a Roll' for breakfast. After all, you're in New York because you want an experience you can't get at home, right? On that note, try to stay away from franchise restaurants for the same reason. Besides, they're too expensive in New York anyway.
Find a place where you can stop and listen to the street musicians playing or smell the great restaurants at night in Times Square. Visit a fresh fish market in Chinatown. Order a bag of sugar-coated coconut from a street vendor.
New York City is safer than it has been in 30 years! The Times Square area has police everywhere and is very well lit, even late at night. If you want to feel the safest, stay away from unlit or uncrowded areas of NYC and put your cash, credit cards and picture ID in your front pockets, instead of a purse or wallet.
Another word on safety: there's lots to see in NYC, but don't get too distracted. If you stay aware of your surroundings and walk swiftly like the natives, you'll do just fine.
Best Way To Get Around:
Try all the different ways to see Manhattan. Walk around to experience the sights, smells and sounds of the City. Take a bus to see that walking is much faster. :-)
If you really want to get around cheaply and quickly, the subway is the best. It isn't hard to figure out the routes (Manhattan is a narrow island) and the daily or weekly passes cover unlimited subway/bus rides and transfers. The subways have places you can stand while waiting so that you will be seen by the employees and undercover cops, if you're concerned about safety.
Also, don't forget to ride the free Staten Island ferry. You can see the Statue of Liberty, but instead of riding crowded tourist ferries, you'll be hanging out with locals as they're traveling to and from home.