Alas the "Desert Wind" has been discontinued... That's the one I took from Chicago, IL to Las Vegas, NV.
You'll actually have to take two trains and it will probably take you at least five days to cross the continent. All of the westbound trains from the East Coast stop in Chicago, with the exception of the JAX-LAX train. For that one you'll have to take a train from New York down the East Coast to JAX.
I agree that the "Empire Builder" is great. Although I have never been on it I think it probably has the most travel time in mountainous territory and I like that.
A nice thing about the trains, as has been already said, is the social opportunities. Gosh, in two days I met people my age from three different countries, ate dinner in the dining car with people old enough to be my grandparents (they had great stories!) and still got to Vegas in a relatively good mood.
Two words of warning, though:
1. AMTRAK does not have a very good on-time record on the western routes. Be prepared to be several hours late -- six was good on the Desert Wind!
2. Do NOT take the Lake Shore Limited from Boston or New York to Chicago... unless it is your only option. It has a terrible on-time record, you are likely to miss your connection if you are more than four hours late to Chicago... That train is also notorious for running out of food and having maintenance problems with toilets, probably because of being so late all the time.
Up, up and away!