The telegram went by steamer Golden Gate to Fortress Monroe, VA. From there, it went by telegraph to the war office next to the white house, from where it was re-transcribed in handwriting, and delivered to Lincoln. Original documents are on the web site.
If you have any further questions after perusing the web site, please let me know.
Um, I don't get something on that link. On the section of when the telegraph reached Lincoln it says
Sherman wrote in his memoirs that: "This message actually reached him on Christmas-eve..." However, in his Civil War News review of Stanley Weintraub’s book General Sherman's Christmas, Carl Schenker Jr. writes: "[Weintraub] puts Sherman’s Christmas gift message into Lincoln’s hands on Christmas Eve, one day early according to the extant military telegram."
Now to me Christmas-eve and Christmas Eve are the same thing, December 24th. Looking over it seems to be suggesting Sherman was saying the night of Christmas, which I'd call Christmas Day eve or Christmas Day night. Is that what Sherman was actually saying or was Christmas Eve also spelled Christmas-eve back in the 1860s?
Thanks for pointing out the inconsistency. I should have quoted the entire comment by Schenker, to show that "Christmas Eve" statement in the book was an error, and that the telegram actually got to Lincoln on Christmas Day. I have revised the Sherman-to-Lincoln Telegram web site and it should be clear now.