I'm a native NYer that commutes from Brooklyn to Midtown Manhattan once or twice a week. I also know most of the city, including the Bronx and Queens, but not Staten Island (is Dat an Island?), and know every area of Brooklyn. I know most of the good bike routes through and around, and know the bad ones too.
I always recommend the West Side Greenway as a good starting point, Prospect Park in Brooklyn, and some of the N/S Avenues in Manhattan. Stay out of Central Park though on weekends, it is a total zoo. The Brooklyn Bridge is also not a good spot to ride with all the tourists on it now, but as a tourist it has become a "must do". As a Brooklynite, I stay off it and use the Manhattan Bridge. The East Side Greenway is a work in progress to be generous, mostly it just isn't a bike trail, just a few connected sidewalks along the river. A fun loop is the Queensboro Bridge, through Williamsburg, and then over either the Manhattan or Williamsburg bridge back to Manhattan. You get to see the hipsters in Billyburg and can stop for a beer somewhere.
If you really want to see the entire city, sign up for the NYC Century this coming Sunday. You don't have to do the whole 100 miles, in fact the extra 25 miles over the 75 mile route is pretty boring stuff through eastern Queens, and maybe the 55 mile route will cover the best parts.
Send me a PM and maybe we can arrange something.