From my personal experiences, my best shots came during sunrise and sunset.
For downtown, I suggest starting at the South Street Seaport/Brooklyn Bridge area during the wee hours of dawn; it is worth getting up that early then maybe going to City Hall around midday (I tried sunset there, threw my metering wayyyy off). Stop by J&R even and see if you need anything

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After that, it's all up to you, but I highly recommend going to Battery Park at about sunset; if you're lucky and you got a good telephoto lens, you can get some great shots of the Statue of Liberty and Ellis Island.
Greenwich Village is quite good also, which includes Union Square and Washington Square just to name a few things; check out Grace Chruch also (again, had metering issues when I went...). The Empire State Building I suggest going to at dusk when the lights start to go on; I had a friend who went up there and took some amazing images. Also, there is B&H around the ESB

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Columbia is also good, which is also by the Cloisters (which I need to go to, considering I have no excuses being a New Yorker all my life).
Then there's Brooklyn and Central Park, but I dunno enough about them to give any suggestions, other than to bring your laptop, lotsa huge memory cards, as much gear as possible and to enjoy yourself.
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Sony P72 (great starter camera), Sony P150 (love the camera, hate the dust), Panasonic FZ-3 (no wonder megazooms are so popular), Canon 30D (ditto for dSLR's)