From Belmond, the company that runs the Orient Express and the Royal Scotsman, comes two luxury trains in Peru. The spectacular Andean Express is considered to be one of the most scenic rail journeys in the world. The second is the Hiram Bingham that goes to Machu Picchu. These are high elevation journeys.
The Andean Express runs from Arequipa (at 2,335m, 7,661ft) to Puno on Lake Titicaca (3,812m, 12,507ft) to Cusco (3,399m, 11,152ft). The train journey can begin from either Arequipa or Cusco. The train eventually reaches a height of 4328 meters (14,200 feet) at is highest point. (They actually carry oxygen onboard in case a passenger is susceptible to altitude sickness.)
The Hiram Bingham runs from Cusco to Machu Picchu (2,400m, 7,874ft).
And there is a Belmond run hotel near Machu Picchu called Sanctuary Lodge.
From Lima, you can fly to either Arequipa or Cusco, or take a bus to Arequipa. Here is a good video of a couple riding the full length of the Andean Express.
