These four trains can carry you from coast to coast, where the final prize awaits: the vast beauty of the West

On the morning of Aug. 12, my wife, Laurel, and I were in New York City’s Penn Station awaiting Amtrak’s Northeast Regional No. 141 to take us to Washington, D.C. This would be the first leg of a 3,447-mile, four-train odyssey to Seattle — in the other Washington — and it would offer a window into...