Taking the subway is going to be your best bet for getting around the city quickly--also a cool place to take pictures.
- You can get a lot of cool shots of the Brooklyn Bridge and Lower Manhattan from the bridge.
- Grand Central Terminal is a nice photo opportunity--although it is usually busy with commuters
- Walk around a few neighborhoods and you're bound to find some spontaneous street shots--Chinatown, Greenwich Village, etc.
- Rockefeller Center: Going to the 'Top of the Rock' is a bit on the pricey side, but there are some amazing skyline shots to be had when you go at night (great views of the city and the Empire State Building).
- Empire State Building: Not much to be said here other than it's the tallest structure in the city.
- Times Square: Most of the shots taken here are a bit cliche, but if you get on one of the islands in the middle of the street you can do some nice long exposures. Generally an NYPD officer will prohibit you from using a tripod though.
- Coney Island: The subway ride is a little on the long side, but there's an amusement park (Astroland), the original Nathan's location, a beach, and all kinds of cool people, like this guy I saw about year ago. He had a live snake about 10-12 feet long and two parrots.
- Liberty City State Park is a ferry ride away for photos of the Statue of Liberty. Living in NYC my entire life, I still have not been inside the Statue of Liberty, so I'm not entirely clear on the situation after 9/11. I think that you can still go up to Lady Liberty's head, but not up her arm to the torch. Don't quote me on that though.
Most of the locations I've named are not too off the beaten path, but like I've said walk around bit and you will find yourself clicking away.
Will you be here for the 4th of July? We generally have some really cool fireworks in the harbor.
Some 'touristy' stores you might want to hit are B&H SuperStore (Bet. 33rd and 34th Streets @ 9th Ave), 5th Avenue Apple Store (59th Street @ 5th Ave).
If you have any other questions about transportation or anything, I'd be happy to answer.