I'll leave the Brooklyn section to folks who live in Brooklyn.

My current route to get from the Manhattan Bridge to the west side is to just continue along Canal Street, across the Bowery, to Elizabeth. Then turn west on Prince Street and from there on it's bike lanes. Follow Prince till it turns into Charlton. Then take a quick right on Greenwich Street, left on Houston for two blocks where you can catch the West Side Greenway.

Traffic gets pretty thick on Elizabeth but I haven't found a better street in that area between the bridges. The upside is it's a through street, with mostly passenger cars, and you don't have to dog leg around the Bowery or ride against traffic.