Though no Civil War battles occurred in New York, the state provided the second largest number of regiments to the Union Army after Pennsylvania. Civil War history happened here and we have a good number of sites. Check out my series about those sites at http://www.civilwaretal.blogspot.com.
There is nothing but a roadside marker in Elmira where the prison was located but I seem to recollect quite a well kept cemetary where many of the confederates who died there are buried. The 64th NY trained there before it was a prison.
I also believe there is a marker in Westfield where the little girl is said to have suggested to Lincoln that he grow a beard.
There are a lot of civil war records, books, and artifacts at the Fenton history center in Jamestown. It is located in Civil War governor Fenton's home which is quite nicely preserved. Well worth a visit.
Funny that subject should come up, only 2 days ago I picked up a book in a flea market called "Final Camping Ground: Civil War Veterans at Brooklyns Green-Wood Cemetery,in Their Own Words" which chronicles excerts from the letters and diaries of men buried there. It also comes with a cd-rom that contains the names, service records and resting place ( plot numbers) of the 3,000 ACW veterans that have so far been discovered at Green-Wood. I have not yet had chance to delve into it but it sure looks interesting.