Whenever I can spare a little time I try to get out for a few hours and shoot for myself around the city. This is how I make a lot of the imagery from my Everyday series, which has basically evolved into a never-ending stream of experimental work. Shooting in such a casual way is a really important part of photography for me for many reasons. Mainly, it forces me to step away from the computer, allows me to try new things without consequence, and it's meditative because it clears my head to let new ideas in. A big city such as New York is the perfect backdrop for this type of work. In NYC, there are many interesting moments happening all day and night, and each neighborhood/borough has unique characteristics, so it never gets old.
The best advice I can offer for shooting in NY (or any dense urban area) is to force yourself to stay in a single location long enough to make at least one interesting image before moving on. That advice sounds really simple and obvious, but many photographers think that shooting great unstaged photographs relies only on being in the right place at the right time. Don't embrace that mentality. There are many ways to make interesting images that don't involve serendipity or something miraculous happening right in front of your lens; you just need to spend some time studying your surroundings.
Learn to see both macro and micro, try looking up, try looking down, pay attention to color, light, symmetry, and scale, shoot a subject through vegetation or objects, hop a fence when no one is looking, ask a stranger if you can shoot his/her portrait, etc. Basically the longer you shoot in one location, the more familiar you and your camera become with it, and as a result, more interesting images can be made.
GEAR OF CHOICE: Canon 5D Mark IIs, a Fuji X-Pro 1 (which I carry everywhere), iPhone, and some polaroids.
With this advice in mind, my biggest warning would be to avoid the "pics or it didn't happen" mindset, which is sort of a tourist mentality that's been infamously inflamed by both social media and ownership of an iPhone or DSLR. We all see far too many posts on Instagram or Facebook from people who snap thoughtless pictures that they'll never look back on, just to tell everyone that they were there. If instead, you learn to take the time to discover new places through your camera, when you review the images you can see the exact moment when you finally started to understand what made that location or moment special. From that point on, the images are always far more interesting and memorable than the first few. So even if you're shooting to post to Facebook or Instagram, spend a little more time on site and try to make a picture that will survive as something meaningful without the accompanying status or geo-tag.
Lastly, always keep a camera on you, have fun, and be patient.