To plan such a trip, your options are limited: For Yellowstone, I believe the most practical advice is to go to Salt Lake City and rent a car. Doable, yes, but a fairly considerable drive, but then you have the car to tour the Park, which is wonderful. You also have a rather large group, involving more than just a regular car. I do highly recommend the evening chuck wagon dinner tour from Roosevelt Lodge on the north side of the park, which takes you from the Lodge in wagons out to a lavish cowboy cookout. Sounds a bit hokey, but the food ( almost 10 years ago ) and the setting were right out of the Old West.
Another idea involving the CZ would be to go to California, and rent a car from say Sacramento, or take an Inland train to Merced for a bus connection to Yosemite, not involving a car.
The only National Park directly accessible by Amtrak is Glacier in Montana, and in that case, a rental vehicle not a necessity. Grand Canyon is accessible by rail by Amtrak to Williams, then Grand Canyon Railway to the Canyon, which is doable, but the connections at Williams are a bit awkward. Hope this helps. I used to work at Grand Canyon - feel free to message me.