Each train will be a little different. Last year it was:
1. Texas Eagle/Sunset Limited from Austin to El Paso. I then bicycled back from El Paso.
2. Bicycled from Abilene to Minot. I then took Empire Builder/Texas Eagle back to Austin.
3. Texas Eagle to Texarkana. Bicycled from Texarkana to Memphis. Rental car to Little Rock. Texas Eagle to Austin.
All these trains now have "roll on/roll off" service. This means I reserved a bike slot and dropped off/retrieved my bicycle from the baggage car. In the past I've also taken other Amtrak trains as well.
There some things I like and some I don't as much for Amtrak and bicycle touring:
Like -
* I have found them to be fairly flexible in making last minute reservations or making some changes to my itinerary. For example, I didn't book my return trip from Minot until I was underway and could more precisely guess when I would get there. I also made some changes to my Arkansas itinerary and was able to do without much fuss.
* It really has been "roll on/roll off". I could ride with loaded panniers to the train station in Austin and then put everything in a duffel bag. Once I reached Texarkana or El Paso, it was less than 15 minutes from time I retrieved my bike until I was on the road. These train stations are also slightly less hectic places than some of the airports I've cycled into and in those cases I needed to assemble/disassemble the bike into a box.
* Prices are comparable or slightly less for one-way travel in coach. If upgrade to sleeper (particular as a single person), you won't save money compared to flying - and long-distance travel will take longer. I've done sleepers some of the time and it is a more comfortable sleep - but I am also able to get an OK sleep in the coach seats.
Don't Like
* In my experience ~80% of the time the trip is on time as measured by "within two hours of schedule". The other ~20-25% of the time it has been off by more hours. So you need to anticipate this and go with the flow. I've had better experience with short hauls (e.g. Cascades) and worse experience with longer hauls (e.g. Empire Builder). Last year my train from Minot to Chicago was 7 hours late. I didn't miss my connection because original plan was to arrive ~3pm in Chicago, stay in a hotel and take the connecting train the next morning. So I arrived @ 10pm and it still all worked out. However, particularly if you are planning a multi-hop you might need to anticipate some contingency time for this type of stuff. // In general I have a slightly different time mindset when traveling by train than by airplane.