Your map has you riding through the Holland Tunnel. Can't do that, legally or with the hope of surviving. Same for many of the other roads your map shows you using, such as I-280 and I-80. Roads marked with red and blue shields and white numbers are called Interstate Highways. Bikes are not permitted in the eastern part of the country. You may ride on many Interstate Highways in the western part of the country, such as in Montana, but by no means all of them.
The only way west out of Manhattan by bike is the George Washington Bridge, which is at the north end of the island. I would avoid Newark and surrounding areas unless someone gives you a trusted route. New Jersey is the most densley populated state in the Unitied States. Newark is the most densley populated part of New Jersey. Not the place you want to wing or ad lib a route.
New York State has a series of signed state bike routes that can take you to, among other places, Buffalo. An interactive map can be found here:https://www.dot.ny.gov/bicycle
I haven't ridden any of them. It's my understanding that their pleasantness can vary.
Another option to Buffalo is this, also with an interactive map:http://www.ptny.org/bikecanal/
Yet another option is to take a short train ride from Manhattan to Seacaucs Junction then a second, longer train ride out to Port Jervis, NY (pretty ride.) From there, ride across the Delaware River to Matamoras, Pennsylvania and pick up signed PA Bike Route L starting with Spur Y1:ftp://ftp.dot.state.pa.us/public/pdf/bikes/state_mapY.pdf
That will take you close to Erie, PA. From Erie, you can follow one of the NY State Routes to Buffalo.
Personally, I would not bother riding out of Manhattan without a good route. There are several train and options, including the aforementioned train connection to Port Jervis and the Metro North line from Grand Central Terminal as far north as Poughkeepsie, New York. You cannot take your bike on during peak hours, but that can be worked around, especially if you start on a Saturday or Sunday.