Road riding in NYC can be an intimidating proposition. Unfortunately, the city isn't the best place to be on the road. But luckily, there are plenty of bike paths and parks available in the city if you can't get out. If you can get across the George Washington Bridge, there are hundreds of great routes just across the Hudson River.
For in-city riding, the Central Park loop is one of the most popular cycling destinations, but if you want something a bit different, head over to Brooklyn and go to Prospect Park (there are also weekly races there in the summer). Both parks will allow you to go fairly fast, although you need to be aware of pedestrians at all times.
Outside of the city, though, are some of the best roads around, from flats to intense hills, tough rides to casual coffee shop jaunts. The best route is to take the Hudson River Greenway just over the George Washington Bridge, taking Route 9 up to Nyack, a fantastic small town north of NYC with a great view of the Hudson from nearly any coffee shop you sit down at to enjoy a mid-ride snack.