The best thing about riding a bike down here is how quickly the cars go away. The Financial District street grid is old and narrow and not much fun on two wheels, but it's only a few blocks wide. Head west and you hit the Hudson River Greenway, a protected path that starts at the Battery and runs the length of the west side. Head east and you're on the South Street waterfront past the Seaport. Both are flat, both are separated from traffic, and they meet each other around the southern tip of the island.
We get asked about bikes a lot, usually some version of "is it actually safe to ride here?" Down here, on the greenways, yes. In midtown traffic, that's a different question. Below Chambers Street you can plan a whole ride that barely touches a car lane.
Three routes below, in order of how much time they take. All of them start within a few minutes of John Street. If you're using Citi Bike, docks are scattered through the neighborhood, including along Water Street, near the Fulton Center, and around the World Trade Center. Check the app for the nearest one with bikes in it before you walk over, because the docks near the office towers empty out fast on weekday mornings.