Rental car would be my first choice, one-way from New York to Bar Harbor. I know Hertz has a location at the Bar Harbor Airport, which is on the mainland between Ellsworth and Bar Harbor. It's a long drive, though, figure a full, long day.
There is regularly scheduled commercial air service to Bar Harbor airport (BHB), but it is small commuter planes and I'm not sure if they have room for bikes or not. They code share with US Airways out of Boston. Jet service is to Bangor airport (BGR) and I believe there are regular shuttles from there to Bar Harbor. I'm not a huge US Airways fan, but they do have some convenient connections to Maine via Philadelphia. You could fly AMS-PHL-BGR and then bike to Bar Harbor/Mt. Desert Island from Bangor, as DaveB suggested. Ride to NYC, spend a few days there, then take the train (AMTRAK) back to Philadelphia and fly home.