Not sure if Amtrak still does this, but a few years back when I used to take it very frequently, on some routes the specific seats were not reserved -- you were guaranteed a seat, but not any particular seat. So if you get on board with four of you, and the train has made previous stops along the route, it is entirely possible that you might not find four seats together anywhere. I would get in touch with Amtrak to ask about that before you go; you don't want to end up with you and one kid in one double seat and the other kids elsewhere. Often people will get up and move but there are always some folks who are traveling solo and just refuse to shift and give up one of a pair of seats. If you let Amtrak know in advance it's very likely they will help you out!
There are, on some cars, seats that are two-facing-two which would be perfect for the four of you. Since you are traveling with young children, I would contact Amtrak now and explain that you must have four seats together (either two-facing-two, or four seats with two on one side of an aisle and two on the other side). Again, this might have changed since I took Amtrak regularly, but it used to be very common on the route I took to see people get on and seek out two or more seats together but be unable to find them, often because of single riders who pretended they didn't hear anyone ask, "Could you please move...."
Of course if your route has truly reserved seats where you have assigned car numbers and seat numbers, you're good to go and all the above is moot....
Power up your batteries on devices -- if there are any plugs, people with devices will jump on those early. Don't depend on getting access to an outlet.
Take non-electronic stuff the kids can do with their hands, like Wiki-Stix (wax-coated heavy thread that kids can shape and reshape into lots of things -- they are fantastic for travel, totally non-messy, can be reused over and over; any good toy store will know what they are), or coloring books, puzzle books for the older child, magnetic or Velcro play sets where the kids can pose characters on backgrounds, etc. Be sure to bring headphones for the electronics if sound is involved!!
Be sure to bring your own food. There are café cars and restaurant cars (two different things) and that's fine, but young kids might not like what they have, and the offerings used to be OK but very limited -- a train can only carry so much. So be sure to have plenty of food you know your kids will eat. A trip to the café car is of course a lot of fun just for the experience!
I always had very good experiences with Amtrak, but I was not traveling with young kids. I do recall one ride of four hours where a mom had two kids with her about the ages of yours and she had nothing, absolutely nothing, with her to occupy them -- not a single book or game or anything was brought out over the whole four hours. I sat facing them so I know! They didn't act up, which was to their great credit, but they looked SO bored. Carry more stuff than you think you need.