Sis ---
You've heard the extremes, as well as the middle ground. The train is just a rolling small town, with people of all stripes. Some will want to interact with the kids, and some will want to keep to themselves. As kids grow up, they learn how to tell the difference. It's part of normal socialization. They shouldn't approach others if you're not around, and they should introduce themselves to adults with their first name only --- no last names. Most people, whether on the train or in the general population, are fine and honorable, but you should use a normal level of caution in the presence of strangers. In the highly unlikely event that you encounter a stranger whose behavior is inappropriate, you need to notify a uniformed Amtrak employee right away. However, it's highly unlikely that this will be an issue. I suggest that you should stick with them, but not keep them on an actual or figurative leash.
There are likely to be other children on the train, especially if you are traveling when school is not in session, so they are likely to find playmates. As mentioned, running is a terrible idea on a moving train, whether the runner is an adult or a child. Amtrak employees don't do it even if they are in a hurry, and they have better "sea legs" than you have. Likewise, the seats, armrests, and luggage racks are not to be used as a jungle gym. There are a few places where the kids might be able to exercise onboard. Sometimes the area behind the last coach seat is occupied by luggage; but if it's open, it probably provides enough space for an 8-year-old to do some pushups or situps. The lower level lounge space may have a bit of floor space if there are no other passengers there. The kids need to avoid intruding on other passengers' space (a courtesy issue) and they need to avoid blocking the aisles (a safety issue). Quiet games, reading, or movies at your coach seat are great. Someone mentioned maps. Nowadays, everybody uses computerized maps; but I believe you might do well to contact the Visitors' Bureau for the States you will be crossing, and get paper maps that the kids can use for reference. I'm very old fashioned, and I truly believe this is a great way to become geographically literate (seemingly a lost art in America).
If there are three of you, you will probably be given a table of your own in the dining car. If there is heavy demand, the fourth seat at your table might be occupied by a single passenger. If that happens, encourage the kids to be sociable, but not pushy. Again, this is a learning opportunity, and it's part of the socialization that is an important part of growing up.
You are traveling in coach as a family, so I imagine the budget is important, to one degree or another. You can save some money by bringing some of your own food. There are some other recent threads that discuss these issues. Dried fruit, such as raisins, are great. They're not too expensive, they're compact, and they don't spoil easily.
This should be a great experience for the kids. During your Chicago layover, you should be able to explore the downtown area, have a nice meal, and maybe visit Grant Park and/or the Art Institute of Chicago on Michigan Avenue. Just walk a couple blocks east on Jackson Street to the el, take the el to the Adams/Wabash station, then walk a block east to Michigan Avenue. The museum will be across the street.
Have a great trip!
Tom