Have you looked up the time frame for going to Las Vegas by way of Amtrak? It's nearly 40 hours...I say drive. You can stop anytime you want, drive at night if there is another driver, stop and have play times at parks and fast food restaurants, pack your own food, sit where you want with your children in their own car seats that they are used to sitting, play movies, etc.... It is about 30 hours to drive it. Most of it can be done while they area asleep.
I think the comfort of your own vehicle out ways any issues. If gasoline really is an issue consider renting a car or van that has better gas mileage.
I am terrified of flying and would explore any option that kept me on the ground.
So it looks like you options are:
Train: 40 hours of dealing with kids
Driving: 30 hours of dealing with kids but control of when you stop, what you do while traveling, stopping to see some sights, have your own vehicle to travel around town, etc
Plane: Hours in the air, trapped, kids may like or hate, food limited, activities limited but shorter time by far.
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S., I did some more research and there isn't a train that goes to Las Vegas. Here is a link to Amtrak routes.
http://www.amtrak.com/pdf/national.pdf
You can go to New Orleans and get on the "City of New Orleans" train at 1:45pm and head up to Chicago which is 19 hours and 15 minutes, 9am, then wait until 3pm and get on the "Southwest Chief" for 34 hours and 24 minutes to get to Kingman Arizona, 11:30 pm, then get a bus or rent a car to go on into Las Veges and you'll have to wait until morning to get rent it because it will be so late.
The return trip consists of getting back to Kingman, returning the car, then waiting until 3am on the "Southwest Chief", get back to Chicago around 3:30pm, then wait until 8pm for the "City of New Orleans" and get into New Orleans about 3:20pm. Here's a link to the trip.
http://tickets.amtrak.com/itd/amtrak/FareFinder?_tripType...
When it comes right down to it you really only have 2 choices unless you really want to spend 108+ hours on the train with layovers of over 10 hours in Chicago. In other words 5 days on a train...and spend over $2000.00 on fare and carry on food and water for snacks and to keep hydrated. Then still rent a vehicle.
Drive or fly, gasoline is much less expensive. Flying is much less time.