We took Amtrak from MSP to Chicago and then rented a car from Chicago to visit family in Indiana.
Taking the train is MUCH better than driving with little ones. We took our 6 month old (at the time). Like the last person said, there is tons of room at your seats and you can take a cooler with you and just about anything else you need. You will need to read Amtrak's policy on carry-on luggage and checked luggage. I would highly suggest getting a cabin for such a LONG trip. You will want to sleep in a bed and you will want your little ones to nap easily. Taking the train will take longer than driving, but can be a lot easier depending on your children's attitude and behavior in a car.
Even if you rent a cabin, you can still sit in the general coach sections while traveling. There is a snack/observation car which is a lot of fun to hang out in. They have tables where families can sit and eat or play games. The snacks are pizza, burgers, etc. that are microwaved and expensive. The dining car is nice but also very expensive. With a family of 4 you will get your own table, otherwise they put people together to fill a table (we shared a table with two other men).
Everyone can go between cars very easily and safely. Every car has a bathroom that is decently sized...but similar to an airplane restroom.
They have water fountains that you can fill a cup with on each of the cars. And at every set of seats (2 on each side), there is an electric plug-in for you to use your laptops or DVD players. They have some seats that face each other so you can sit across from your family.
When you board, you get to sit where you want to and move around if you choose. You will just have to take your paper tickets that they give you and place overhead of you with you (if you move).
You may really want to compare prices. Now with Southwest flying out of MSP, it is much cheaper to fly. If you fly, take the car seats with you on the flight and put the kiddos in them.
If you take the train and plan to rent a car when you get there, I'd highly recommend "renting" the carseats from the car rental places. There is no need to take them on the train.
If you have any more questions about Amtrak or flying, let me know. We've done it more times than I would have liked to!