The biggest problem with lightspeed transportation would not arise while travelling at full speed, but accelerating. Acceleration is any change in velocity, even though it usually refers to speed. The biggest changes in acceleration we'd need to worry about would be departure and arrival.
Moving at high speeds through an atmosphere is difficult due to a little something called air friction. It is because of atmosphere that we have supersonic speed at all. Something many people do not understand is that sonic booms are not a dramatic event that occurs when a thing breaks the sound barrier. They trail behind it until that thing returns to subsonic speeds. The reason we do not have supersonic ground vehicles is that sonic booms can be highly destructive. Until you got that hyper-train up to near-light speed, you'd need a way to contain the sonic boom until you can just phase through the air.
There is also the problem of not only creating an machine that can move at light speed, but harnessing it. How would we contain such a thing? How would we attach it to the chassis? How would we keep the train car from dropping its occupants and speeding off without them? How would we even load the passengers onto the train?
I haven't even started on the subject of brakes. How on earth would such a thing stop? How would it stop without killing everyone on board? How would it stop in such a way that we could reuse the engine and vehicle?
What of the passengers? What will they experience at near-light speeds? As Hans said, things get weird when you go that fast. How would you retain normality travelling through what is almost the fourth dimension? What would the passengers experience? Would they ever be the same again? Would they eventually be able to accept it as normal or would everyone who rides the machine be cursed with some form of madness contracted by seeing aspects of reality humans were never meant to see?
That would actually make a great premise for a horror story. I'll have to remember it.