Fares are based on segments. A segment is generally described as anytime you get off the train then back on a different train whether it is the same train number (same route, different day) or a different train number (a required change, ie: going through Chicago).
If I were to book a reservation from San Diego to Seattle, it would most likely be the same price if I booked two separate reservations - one from San Diego to LA and another one from LA to Seattle. However, if I were to book from LA to Seattle, it would be cheaper than booking LA to Portland and another reservation from Portland to Seattle.
There are always exceptions, but those are the general rules. This is further recognized by the fact that they offer a segment-based rail pass.
I hate to disagree with johnny, but when you book a reservation, you book each segment for a date and train number. If you choose to get off during that ride, there is no obligation for Amtrak to carry you the rest of the way that is not covered by your ticket. Most trains are "reserved". Who is to say that if you were on Saturday's Coast Starlight and got off somewhere along the route that Sunday's train isn't sold out?
There are ways to manipulate the system through creative use of connections to get time to spend at major terminals, but it is almost impossible to get off/on priviledges for a single segment fare.
Some short distance routes, like the Pacific Surfliner - but NOT long distance routes like the Coast Starlight - offer monthly passes, but it is unclear if they are even permitted for on/off priviledges anywhere enroute or if they are restricted to the endpoints.
Finally, as I eluded to above, there is the USA Rail Pass which allows limited off/on priviledges constrained by segment limits. And California has their own rail pass which offers 7 days for $159 but only within California (any point beyond CA is a full segment charge). Texas Eagle, Sunset Limited, California Zephyr and Southwest Chief don't participate.