Trains can and do indeed have problems in the snow; it really depends on a few things. How fast the snow falls, how deep is the snow, and how cold is it.
A sudden rapid snowfall over 10-12 inches within a few hours can often cause problems, as the crews can’t clear the tracks fast enough. When the snow falls at a slower rate, the crews can keep up with plowing and clearing switches, plus the normal passage of trains also helps to keep the tracks clear.
As the depth of the snowfall increases it also gets harder for the crews to keep up, not to mention that you have no place to push and/or blow the snow to. So again this can cause a train to be canceled, delayed, and/or rerouted.
Finally the temperature can play a big roll, as the closer the temperature gets to freezing the wetter the snow gets. Wet snow tends to be more slippery and harder to move first of all. Secondly if the temperature falls later on it can lead to freezing on the switches. This prevents the switch from moving. Of course even loose snow packed in between the points of the switch can hamper it’s proper movement, however ice is even harder to clear than is plain snow.
All railroads actually have a rating system in place to determine the severity of the snowfall and it’s impact on operations. Each level determines what trains are allowed to move and which trains are sidetracked. At the highest level, only perishables and passenger operations are allowed to keep moving. Even then they attempt to limit the amount of switching as much as possible from one track to another, and trains can still be annulled and/or stranded if conditions deteriorate further. Those levels also determine at what point the railroad invokes Force Majure, which cancels all on time guarantees.
Amtrak does have a pretty good record in the snow, but they have been forced to cancel trains in the past. Amtrak's record of getting through however, is far superior to that of the Airlines.
Hopefully the snowfall that you see will be light and just enhance the scenery, without interfering with operations.