L.R.
J., I've traveled by train a lot and wouldn't even do this long a trip--especially with no sleeper car--with my daughter, who is seven and a very easy, experienced train and plane traveler. The thought of doing it with three kids and one of them only 2 is too much. I'd fly if at all possible.
You will be lugging a LOT of things to keep them busy and a LOT of food -- the people below are right, train food is either very pricey (on the diner cars where you sit and order, and it's limited selection) or very unhealthy and limited (in the "cafe" cars where the limited food sells out very fast and you may be faced with offering your two-year-old nothing but danishes and soda). Do you want to haul lots of food, lots of toys, lots of cleaning stuff? (Yes, the toilets get very messy and are unisex, so the floors and toilets get wet from men using them while the train's lurching--sorry if that's gross, but it's true).I have taken Amtrak a lot too and don't know about the thing mentioned below where seats can be "turned" to face each other -- I have never seen that happen but it may be available. There are seats that are permanently built-in to face each other at the ends of some (not all) cars but if someone else is in those when you get on, you're out of luck. At least here on the East Coast line, Amtrak regular coach seats are not reserved - you get on the train and take whatever isn't already taken by passengers who were on the train ahead of you; you may have a tough time getting four seats together at all, and do you want your kids sitting behind you or even several seats away where you can't see them? And finally, at some point, the 2-year-old will want to run up and down the aisles to blow off steam. Can't blame her. But it isn't safe because trains do bump and rock, and it will draw a lot of attention you may not want.
If it were just the 8 and 6 year olds and they are very good at occupying themselves and OK with sleeping sitting up, you might manage OK, but especially with a toddler along, I'd say, fly. Don't get me wrong, I love train travel despite what I'm saying here! But for very long hauls, with small kids, I've witnessed a fair number of family meltdowns.