Going by memory here as someone who lived in upper Manhattan and would spot Empire Service trains in the neighborhood...
The Adirondack (and all Empire Service trains) crosses into Manhattan from the Bronx on the Spuytin Duyvil bridge, and stays on the west side of the island near the Hudson River for most of the trip. Heading south, the train passes the lights of the Columbia University football field (as it crosses the bridge), and runs at or near the water's edge (sometimes separated by a park or highway) past the Dyckman soccer fields, under the George Washington Bridge, and along Riverside Drive. From about 125th Street south, the train operates in a tunnel under Riverside Park, roughly level with the West Side Highway--there's occasionally some light through grates on the highway side. South of 59th/72nd? there are some tunnels and open cut sections to reach Penn Station.
A good map will show the NYC/Amtrak tracks along the Hudson River. It's an unusual stretch since it was originally built for freight trains and is not double-tracked the whole way.