Making a reservation and taking a trip by Amtrak are two different things.
Making a reservation can be frustrating when it includes the requirement of taking a bike. The reason for this is that Amtrak doesn't have many stations that provide baggage handling and baggage handling is required to load and unload a bike from the train. As has been mentioned your best resource for narrowing this down is the Amtrak website where the stations with baggage service are shown. Amtrak is well-known for providing different answers to the same question when you ask more than one Amtrak employee.
The actual trip can be quite nice depending on a number of things; it is probably not fair to readers or to Amtrak to generalize about this. Each trip is different. Sometimes things can go wrong; sometimes trains in the US are late or behind schedule. In that sense it is like bike touring in general.
Bicycle touring requires a personal, individual resourcefulness balanced with a willingness to go with what is handed you at the time. In some cases you can seek other options but in many you can't.
When something does go wrong some people look for someone to help them, others try to rely on their own resources without seeking the help of another person and some people just tough it out and don't much look for help or alternatives. Each of those approaches is valid. You will know what your natural style of response is when you get out there and something does go wrong.
Remember what Yvonne Chiounard said, "It isn't really an adventure until something goes wrong."