I'd suggest you start with reading the Seat61 page on Train Travel in the USA. That'll tell you about the main routes, the main trains, general timings and prices etc.
As a general rule, most big places on the east and west coast can be visited, but only certain ones inland (in other cases the appropriate lines have closed). For long distances, the trains are normally slower than an equivalent european one, and don't run as frequently.
There are a few areas with the kinds of frequency and speed you get in Europe, but they tend to be along the east coast (such as Boston - New York - DC)
For information and geographical accuracy, you probably want the Seat61 Map, but the official Amtrak one will give you an idea of roughly where you can travel: