New York to Seattle, leaving February 12, 1906:
That was at the time that the US railroad system was approaching maximum complexity.
I will assume through Chicago. That would be about 95% most likely. Going through St. Louis would be most of the remaining 5%. I would presume he was trying to travel as cheaply as possible, so if he could find a cheap ticket by a roundabout routing, it could be through a few other places as well.
Did he make it as one trip? Did he do it as a passenger or by hopping freight trains?
If he came as a more or less typical immigrant of the times, he may have made the trip is several small pieces, stopping to work a while to make money to go further. Any information you could provide about points he went through would be helpful.
New york to Chicagl:
At that time, probably 50% of rail travel went by way of New york Central, Chicago, Albany, Buffalo, Cleveland, Chicago. Someof the NYC trains went across southern Canada, in which case he would have gone through Niagra Falls and Detroit.
Most of the rest of the rail travel probably went by way of teh Pennsylvania Railroad, in which case he would have gone through Philadelphia and Pittsburg.
A smaller portion went by the Erie Railroad, which sort of split the difference between the routes of the NYC and Pennsy. He could have possible also went by way of the Baltimore and Ohia which would have sent hime through Baltimore, and possible Washington DC, and then west through Pittsburg. He could have also taken either the Lackawanna or Lehigh Valley to Buffalo and then taken either the Nickel Plate or NYC west to Chicago.
After Chicago, it would be simpler. The Milwaukee Railroad's Pacific Extension had not yet been built, so a one railroad company ride west from Chicago was not possible. Most likely his next point after Chicago would be St. Paul, Minnesota. There were three relatively equal possibilities for that connection. Since both Great Northern and Northern Pacific had a relationship to Chicago Burlingto and Quincy, later, if not at that time. CB&Q would be the most likely, in which case he would not go through Milwaukee. If he went through Milwaukee, tne line would be most likely either Milwaukee Railroad (CM&St.P) or NorthWestern (C&NW). There may have been one more company later absorbed by C&NW, but I do not know the history of that area that well. Less Likely, but possible would be to take the Rock Island (CRIP). From St. Paul, there were only two possibilities to Seattle, the Great Norther, which is the current Amtrak Empire Builder ourte, or the Northern Pacific, which went through southern Montana. If the target was Fairbanks, did he go overland from Anchorage?