Right now there are two trains from Grand Central to Manitou and Breakneck Ridge, on weekends only: 7:51, arriving 8:59 and 9:12, and 8:51, arriving 9:59 and 10:12. Return trains leave at 4:55 and 5:08, arriving at Grand Central at 6:26. The later train leaves Breakneck at 6:55, Manitou at 7:08, and arrives at Grand Central at 8:26. These times will change slightly in the summer, not more than five or ten minutes.
The Breakneck stop leaves you across the street from several trails into the Hudson Highlands park. The Manitou stop leaves you about a mile from Anthony's Nose, the Appalachian Trail, and the Bear Mountain Bridge.
There is also a Harlem line train from Grand Central to the Appalachian Trail near Pawling, but I don't have the times for that.
The Short Line runs a bus from the Port Authority to Bear Mountain Inn, departing 8:45 and arriving 10:15. Return buses are at 3:19 and 5:19. The Short Line also has other stops near Harriman. This is a very slow, uncomfortable, and expensive way to go.