The ONLY train system in the U.S. is Amtrak. See map below to see that it is a very long trip from Houston to Las Vegas by train. It is also NOT cheap. You could probably get plane tickets cheaper. The Amtrak site is kinda hard to use, so if you want to, you can call and ask about the number of hours for that tip and the fare.1-800-872-7245
There are no trains that go from Houston to Vegas. Houston does not really have much of a train system. The only train in the city used by the public is the light rail that flows from downtown. You could try getting there by bus, but to be honest, it would be about the same to fly as it would to go by bus.
You re better off driving which is far cheaper but takes two days (and you have to get a hotel room in El Paso or so). It is also cheaper to fly round trip.