You definitely do NOT want to ride over the Manhattan Bridge - I don't believe there is a bike path. [IMG]/img/vbsmilies/smilies/biggrin.gif[/IMG]
The Brooklyn Bridge has a bike path but if the weather is nice it's best to start early so as to avoid tourist/pedestrian congestion.
I don't know of a bike path through the city (i'm sure there are though), but there is a bike path along the West Side Highway, which goes all the way to the George Washington Bridge. Central Park starts at 59th Street so you should be able to cut across Manhattan at any appropriate street after that. Not sure of any specific bike path that goes all the way, but these days, much to the angst of pedestrians, motorists, and cylists alike, they are all over the place. Maybe the rental shop has a map or some recommendation? I don't find it a prob as depending on the circumstances sometimes I actually find it safer to ride w/traffic bike messenger style (with a good dose of politeness to any pedestrian traffic along the way).
Central Park will be VERY congested on a nice day, IMO unrideable on weekend afternoons. You may have a better ride going up the west side highway bike path, I'm not a professional travel guide though. If you are in Brooklyn, Prospect Park is also a lovely ride. A little short at 3.5 miles but if don't mind being a hamster on a wheel, 10 laps works out to 35 miles. I do that 4X a week. Good luck and enjoy the Big Apple!