The first few posts explained the majority of the problem very well. But there's a major piece of history that also has to be taken into account. The result of this history is that US railroads are for the most part privately owned and operated. Yes, in a few areas the road itself is owned and operated by Amtrak, and in a few areas by state gov't - NC and NM come to mind (also MI?). And there is an increasing tendency for gov't to buy out the private company and run and maintain the road itself. But the vast majority of the roads are still privately owned.
Contrast this with virtually every other country in the world, where the rail-road (infrastructure) is gov't owned and private companies pay to run trains on that road. The Eurozone is undergoing a revolution in rail operation, the whole point of which is to open access to all the gov't owned railroads to any and all train operators. So far, it's working beautifully. Operators, both freight and passenger, are competing with each other and, as best I can tell, prices are going down while service is going up. This is the way we do highways (with a few exceptions) and air routes and waterways in the US but not how we do railroads. And until this changes, we're stuck with slow infrequent passenger service no matter how much money we throw at it.
Yes, gov't can build brand-new high-speed lines. Buy the land, and build the track, and then either run or franchise the operations or just possibly allow multiple companies to compete, though this becomes harder and harder as speeds increase and the necessary technology gets more complicated. But, as others keep saying, HSR really only makes sense along the densely populated corridors, which still leaves a huge part of the country in need of much better passenger rail. And even in the corridors, it would be insane to build HSR w/o building on existing freight corridors. No, freight and HSR can’t share track but they can certainly share right-of-way and often the land adjoining the existing r-o-w’s would be cheaply acquired.
I’m not saying this will ever happen in the US – climate change may preclude any possibility of this before we come to our senses and start the process. I’m only saying it’s a necessary step if we ever are to have passenger rail like the rest of the developed world.
US trains taken: Starlight, Zephyr (through both Wyo and Utah/Colorado), SW Chief, Sunset, Southern Crescent, Cardinal, Carolinian, NEC Boston to Washington (though not yet on Accela), EB, NorthCoastHiawatha, Desert Wind, DRG through Colorado, Silver Star (NYP/CRN), Night Owl, Capital Limited, Wolverine, Sounder, probably others.
Canada, Europe (including USSR), Mexico, Central America, Northern Africa,and Japan.