Harry Beck designed a diagrammatic map of the London Underground back in 1931. But the Metropolitan Transportation Authority sticks doggedly to this approximately geographical format. Points to note: it has been rotated about 30° anti-clockwise from north-up so Manhattan is straight up and down and Central Park looks unnaturally wide because of stretching needed to show the detail in Midtown.
I also found this even more complicated map dated 2010 June in their archive.

Getting about NYC is generally quite easy with their boring grid system of streets and avenues. But one trap to catch the unwary is that you can have separate subway stations with the same name. There are three stations called 103rd Street in Manhattan at least a kilometre apart from each other and another in Queens. This is 103rd Street served by the train.
Those parts of the subway which are actually underground were mainly built by cut-and-cover. Here we are looking up at the sky. Up top these are just metal grilles in the cetral reservation of Broadway.

The guard rides in a tiny cubicle midway down the train. These monitors, I think at 103rd Street for the train, enable them to check the state of the closing doors

Monster vacuum cleaner! Do we have these on the London Underground?

Automated display on the train. Lights on to show where it has yet to stop - important because the <> train does not stop at certain stations.

South Ferry station is the downtown terminus of the train. It is built on a very tight curve - you can only get out from the first few cars and they have these extending platforms to cover the gap. These three imaages show the platform extended - note two yellow strips and …

here is the platform retracted.

An hallmark of the New York subway is the lines of pillars close to the platform edge arising from they way they do cut-and-cover. This is Fulton Street station on the and lines.

An exception to the cut-and-cover construction had to be used at Washington Heights which include the highest natural point in Manhattan at 81 metres (265 feet) above sea level. So the platforms here at 181st Street station on the tunnelling construction was used. The line is reached by "three massive elevators" and and we have a wide space free of pillars.

Clearly the subway builders were not familiar with deep tube building techniques. There was a fatal accident during construction in 1903 and in 2009 an eight metre long section of the brick lining collapsed.

And even here, if we look in the opposite direction along the platform, they have reverted to their beloved pillars.

But by the time we get to Dyckman Street the subway has become overground.

World Trade Center 1 lurking behind other buildings seen from Battery Park

Emma Lazarus wrote The New Colossus, a sonnet which appears on a plaque in the pedestal of the Statue of Liberty. It includes the words: Give me your tired, your poor, Your huddled masses yearning to breathe free, The wretched refuse of your teeming shore.
To get us in the mood, they force us into an huddled mass waiting for the security screening which is almost as strict as airport security.

Picasa's pathetic attempt at stitching together four images of downtown Manhattan

Windows Live Photo Gallery's superb bit of stitching. The new One World Trade Center is in the center.

These crude dog outlines at the foot of the pedestal are pivoted to move about in the wind and in fact are very effective at scaring away the seagulls

This rather sweet mini fire truck is stationed within the terminal because ceiling heights especially down at the platforms would not allow a regular truck in.

One of the few sensible things that Americans do is talk about "track" numbers not platform numbers.

++++ use Draft:Steves & Sons. Link to more similar from previous US trip.

Dump no waste - drains to waterways, ie. to the Hudson. See message cast into the plate at the bottom of the photo. Compare with "no dumping" photos in other parts of the USA.