AFAIK, they normally just turn around the train (physically) to point in the other direction, clean it and restock it and use it for the next trip the other way. They only break it up if there is a problem with broken equipment that can't be repaired between trips, or for some sorts of scheduled maintenance, or if they are adding or removing cars for seasonal capacity changes.
Some regional trains have an engine at one end and a "cab car" at the other. The cab car is usually an ordinary coach with an engineer's cabin at the end. This is very common with commuter railroads, but Amtrak has some routes that work the same way. Instead of turning the train physically, the engineer just walks to the other end of the train and takes control from there.
The process of setting up a train to run the next trip in the opposite direction is called "turning the train around" even when the train isn't physically turned around, because it can run backwards.
Usually, a train from point A to B will be train N in that direction and train N+1 in the other direction. The only exception I know of is the Coast Starlight, where the northbound train (LAX to Seattle) is train 14 and the other direction is train 11.
If the trip (including the turn-around time) is longer than 24 hours, they will have more than one set of equipment and will rotate between them, so there can easily be, for example, two train 7's running at any given moment. If it is necessary to distinguish them, they are labelled with the departure date, e.g. 7(16) and 7(17).
As far as I know, Amtrak doesn't have any 3 point trains, where a physical train runs as train 1 from point A to Point B, then runs as train 3 from B to C, then as train 5 from C back to A, while a second train runs in the other direction as train 6 from A to C, then as train 4 from C to B and finally as train 2 from B to A. (Or as 1 from A to B, 3 from B to C, then 2 from C to B and finally as