This time of year, with the train not full, you should be able to go to the Sightseer Lounge first thing in the morning and find seats for six, no problem! You will find that half of the upper level is made up of tables with bench seats (made to seat four, but if your children are small, your family may fit in one booth), The tables are good for families, because you can spread out books and games to occupy the little ones. They are also less in-demand than the couches and chairs, so you need not feel like you are supposed to cede the ground to someone else after a short time. Your coach seats are reserved for you, so they will be there to return to if someone needs some down-time.
You may bring your own food into the upper level (NOT downstairs where the cafe is, but if you wish, you can also buy hot chocolate, bagels, etc downstairs and carry it upstairs to have with your own food). I expect your children will be well-behaved and enthusiastic about the experience (I went to the U in SLC, and was impressed by Utah children), and they will be smiled at with a great deal of tolerance by train staff and other passengers.
The scenery on the section you have chosen is absolutely stunning, and I just know you will have a wonderful experience. I hope you will join this forum and report back on it afterwards.
I'm a "little Old Lady in Tennis Shoes" (LOLITS) from Eastern Oregon. I love to travel by train, though I live way out in the toolies, far from the nearest Amtrak station (Chemult). My station would have been Baker City, but they cancelled the Oregon Pioneer just before I took my first long-distance train trip as an adult. I've taken most trains in the West, but I'm still exploring new routes in the east.