A hotel room inside NY City limits is going to run anywhere from 125 - 200 a night. I'm assuming you don't have your hearts set on staying at the Ritz-Carlton or at the Waldorf Astoria, which are a bit more pricy.
All I ever expect from a hotel is a safe, clean place to crash. If you don't use the fancy restaurant, pool, sauna, and weight room, why pay for it? IME
, you can never go wrong with discount chains like Super 8, Days Inn, Quality Inn, and Holiday Inn.
Look for hotels in Northern New Jersey (Bergen, Secaucus, Union) and Queens (Long Island City, Astoria, Elmhurst) - both locations are very close to Midtown Manhattan, and are relatively inexpensive (75 - 125 per night).
I wouldn't even think about renting a car. It's very expensive, and parking is always a time consuming pain. Besides, the NYC public transportation system is safe and convenient. There are very few places in this city where you're more than a 10 minute walk from a bus, subway, or light rail stop.
You can get system maps here:
A 7 day unlimited use metrocard costs $24 - it's good on all NYC subways and buses.
So, what to so when you get here:
At a kiosk smack in the middle of Times Square, you can get halfprice tickets to same day Broadway shows (depending on availability)
Radio City Music Hall, Rockefeller Center, St Patrick's Cathedral, and Grand Central Terminal are nearby.
The Metropolitan Museum of Art is a must - the Guggenheim isn't far away.
The Cloisters - The Met's collection of medieval art displayed within a beautiful castle and garden. This is quite a subway ride - all the way to the northern tip of Manhattan, but well worth it IMO
. I think you can get a discount pass if you plan to visit both the Met and the Cloisters.
If you like art, stroll the streets of SOHO and Greenwich Village and check out the art galleries. Then have lunch or dinner in Little Italy or Chinatown.
Central Park is a must as well. The newly renovated Great Lawn, Belvedere Castle, and Turtle pond are a stone's throw from the Museum of Natural History, which itself is a must-see. You can rent boats and go rowing on the Lake too.
Empire State Building - a must.
Statue of Liberty - worth the trip only if they're letting people into the statue, otherwise, you'd be better off doing the CircleLine cruise around Manhattan. The Staten Island Ferry is one of the few remaining freebies left in the city. From it, you can get some amazing pics of the Statue of Liberty and what's left of our skyline after 9-11.
Trump Tower is neat to visit around Xmas, but there's really not much to see there except for the outrageously-priced boutiques.
I agree with BrianNY that the South Street Seaport is worth a visit - especially Pier 17. You can get some great pics of the Brooklyn Bridge from the Fulton Street fish market area. Wall Street is actually unimpressive, but you probably want to get your picture near the bull statue.
The Intrepid air-sea museum on the West side is neat if you're into WWII naval and aerial history.
I am very familiar with Manhattan, Queens, and parts of Brooklyn. JChillin is familiar with the Bronx and Manhattan. I'm sure that we can answer any questions that come up as you plan this trip. Have fun!