Well, it depends what your interests are. For a first timer, I'd assume most of what you'd want to see is on Manhattan Island. You can get a place to stay just outside of the city and use public transportation to get around - basically, around here there's nowhere you can't get with a bus or train.
NYC is a lot safer than it used to be, provided you use your head. I'd think some of the stuff a first-timer would want to see would be the Statue of Liberty, which is off of Manhattan's Southwest end. If you head down to the financial district, which is where Wall Street (and the Trade Center/Ground Zero site) is located, you can see a lot of those sites, and either see, or pick up a ferry, to Liberty Island. (Battery Park is down there too.) South Street Seaport and area are great to see in warmer weather - not as much fun in the cold - but once warmer weather hits, there's always tons of stuff to see and do - street festivals, etc.
The heart of the city is in Midtown - Times Square (where 42nd Street meets Broadway) - that's only a few blocks away from Fifth Avenue, where you can see 30 Rock, Radio City Music Hall, St. Patrick's Cathedral, etc. Madison Square Garden, the Empire State Building, and Grand Central Station are all only a bit south of there, within walking distance, and Central Park and a number of big museums are about twenty blocks north - easy walking distance on a nice day, but if you're coming out in chilly weather, better to take a cab or the subway.
There are a clump of neighborhoods between Downtown and Midtown - Greenwich Village, Little Italy, Chinatown, Soho, Tribeca, etc, which more or less all abut each other. There are other areas - Chelsea, Washington Heights, Harlem, that have their own identities but are in separate directions.
Long story short, your question's a bit broad. My best advice would be that the biggest draws are in Midtown and Downtown. Many sites are free to see, but the museums et al. will likely ask for 'donations' for admittance. Getting around is easy - it's actually really easy to get around without a car (given the massive traffic we tend to experience) - a subway map is a good friend, but the buses go everywhere, too.
A lot will depend on your budget, too. It's probably cheaper to stay in a hotel in NJ and commute in - I'd imagine the city hotels are probably pricier, but you'd have to shop around. (I live in NJ, so I don't have a real sense of hotel space in the city.) But once you're in the metro area, getting around is not tough. If you have specific questions, please let me know - I'd be happy to offer what advice I can!