First, the inequities against Amtrak via other methods of travel:
1. The airlines don't have to pay for the airports. They don't pay for them to be built and they don't pay for the maintenance. Generally, the local county or city government covers all of these expenses. The airports simply lease gates from the government authority, generally at a substantially sub-market rate. <i.e the square footage cost compared to a retail shop.> Airports are kept solvent by the retail shops inside the airport.
2. Congress and the President consistently complain about spending $1.5 billon on Amtrak while thinking nothing of spending $800 billion on a national highway bill. Those tractor trailers that ship you your goods are basically getting a free ride courtesy of Uncle Sam. Sure, there is a gas tax, but that hardly makes up for the highway mantainence costs.
3. The freight railroads own and maintain all their own track <with exceptions in the north east and west coast>. As such, they have a say over who runs on their rails. They make much more profit running their own frieght over their lines at 45mph than they do allowing an Amtrak train 60, 90, or 100mph access over the same line.
4. Amtrak owns and maintains all of it's own stations and much of the passenger track in the northeast. This is the biggest source of inequity. If the airlines had to pay for and maintain the airports they use, you'd see air travel in this country change dramatically. Air travel would suddenly become a huge luxury rather than a necessity. Additionally, all the track that runs from DC to Boston and west to Harrisburg is electrified. This is very expensive to maintain, but allows for very fast, clean transportation.
My Solutions:
Amtrak needs to be broken up into two pieces.
The first piece is the infrastructure. A Federal government agency, advised by Amtrak and some of the freight carriers, that owns and maintains all of the passenger rail infrastructure. This agency would also be responsible for expanding the passenger-only rail network much in the same way the Federal Highway Administration does with highways today.
The other piece is a privatized passenger rail entity... the Amtrak that you know today but only the trains themselves. They own/lease the actual equipment such as engines and cars. Their job is to run trains and nothing else. They lease station space from the Federal Rail Infrastructure agency.
The 3rd thing that needs to happen is that an incentive needs to be offered to the freight lines to expedite passenger trains using their rails. Some current freight lines can't handle high speed trains, but even cruising along at 70mph is still better than being stuck behind a coal train barely pushing 35mph.
4th, offer incentives of some sort for the freight rail companies to begin offering Passenger service again to spur some competition. They can lease station space from the Federal Rail Infrastructure Agency just as Amtrak can. The side effect of this, if effective, is that the freight lines will begin to upgrade some of their track on their own.
We will never see rail travel like they have in Europe, but we can substantially improve what we currently have.