Once you get to Switzerland you can buy a rail pass quite cheaply which gives you free travel for a week, or more if you pay more, of course. It comes in standard and first class variants.
The pass is valid on all major railways (a fair mileage is metre gauge rack railway) with a discount on most private mountain railways. You also get free travel on the lake steamers and on the post buses (which drive over some hair raising mountain passes).
Unlike Amtrak, Swiss trains run accurately to the second - they even pass trains non-stop both ways at passing loops, which is hair raising when you think about it.
Bob Gartside