You say you have no idea of the layout of the city.
Here is a simple map that may help
http://www.usatourist.com/maps/nycmap.gif
New York City is made up of five main areas - Bronx, Queens, Brooklyn, Manhatten and Staten Island.
You can see where the 3 NY airports are on this map.
To be honest MOST of the things that the tourist wants to see are on Manhatten.
As you can see Manhatten is an island and MOST of the tourist things are in the South of the island.
Central Park is right in the middle of Manhatten.
The South of the Island includes Empire State Building, Twin Towers (as were), Times Square, Broadway, most of the shops, clubs etc.
The Status of Liberty is off the South of the Island.
But of course the South is the expensive part.
The North of Manhatten (above Central Park) includes Harlem which is an area worth avoiding (at least it was when I went through it by car about 8 years ago)
Here is a map of just Manhatten, and you can see Central Park right in the middle.
http://www.areahotelguide.com/new_york_city_ma ps_civ.gif
Manhatten is about 13 miles long and just over 2 miles wide. But most of the bits you will want to see are in the bottom 3 or 4 miles of it.
My guess is that most tourists spend ALL of their time on Manhatten and only go off it to get to the airport.
Try to get a hotel in midtown or downtown Manhatten.