Road running is probably the easiest sport to get into in NYC, mainly because there's so much pavement, everywhere. Still, running out of your apartment and down Fifth Avenue isn't much fun (lots of tourists and businessmen to dodge), but there are a lot of great in-city options.
If you love having people around, Central Park or the Hudson River Greenway are easy, nearby options with smooth, comfortable running surfaces and plenty of people to pace you. If you're bored of the Manhattan scene, run over the Brooklyn Bridge and into Brooklyn to run the loop at Prospect Park for that similar Central Park feel but with a new group of people and slightly different sights to see.
If you want a bit more solitude, Riverside Park runs along the Hudson in uptown Manhattan and while you won't be alone, it's a bit less crowded.
For some contemplation or a recovery run, the NYC Botanical Gardens can't be beat, especially when in full bloom in the summer. Gorgeous, but not for those with serious allergies!