An autotrain is a push-pull train incorporating a steam locomotive and specially fitted passenger coaches.
Yes, you can ship by private carrier or Amtrak calls it the 'AutoTrain', where you can have your car with you as you travel.
If you mean passenger rail, Amtrak is the only national provider of passenger service. Despite attempts by various political groups to put it out of business, it is VERY popular with travellers. Some services such as the AutoTrain are booked months in advance, and in the Northeast Corridor it provides more intercity service than any other public carrier, including bus and air. If you mean freight rail, the largest systems would include CSX, Burlington Northern Santa Fe, Union Pacific, and a few others. You can look at www.railroad.net for more information, among many sites devoted to railroads.