The problem with doing the US in 1 trip is that it is so big. Sort of like trying to do Europe in one trip. Northern Germany is so different that Southern Spain and eastern France is so different that Austria.
If you mean you want a "western" (as in the movies with mountains, wooden store fronts, rivers, etc.) American feel then I would fly into Portland and ride to Jackson, WY.
If you mean a "small town, friendly (but few) people with mostly wide open views" then I would go from Denver to say Chicago or further east OR possibly Chicago to Jackson, WY if you want some mountain scenery too. Not nearly as pretty (except in western Wyoming) but then you will get a totally different feel as to the western or eastern US.
If you want "older, much more established, busier, more populated" America, then maybe go westbound from Washington, DC, to say Chicago. And Washington is nothing like Wyoming.
I would encourage you to look at the TransAmerica Route journals over on CrazyGuyonabike.com and look at what areas appeal to you so you can give us a better idea. Then we can narrow where you fly into and out of.
For instance, if you go to Yellowstone, you could always take a fairly cheap shuttle from Jackson to Salt Lake City where the airfares should be a lot cheaper. We understand the complicated Amtrak better than you. Sort of like I need a German to figure out the German train website.
Best, John