This, there is NO reason to be going into Manhattan at all, unless you are making a pit stop there. New York State is a lot bigger than just Manhattan. Plus, ALL entrances into Manhattan have a $13+ toll to get in.
An alternate route to the one above (which is how I'd do it,) is to take the NJ Turnpike all the way to the Palisades Parkway, OR the Garden State Parkway to the NYS Thruway, north. Once you are on the Thruway, you have SEVERAL ways to cross the Hudson River, depending how far up the Thruway you go.
You can cross at the Tappan Zee, as listed above.
OR: Take the NYS Thruway north to Interstate 84 East at Newburgh, Exit 17, across the Newburgh Beacon Bridge, skipping all the downstate city traffic, then head towards Connecticut and/or MA.
OR Take the thruway up to the Massachusetts Turnpike. I think it's around exit 22(?) to go through MA. Although, you'd be backtracking back down a bit to get to the Mass Turnpike. But the Thruway is very speedy, unlike some of the windy CT highways.