Most of Amtrak's trains allow passengers to reserve a seat by booking ahead of time. As with flights, try to reserve your spot as early as possible -- particularly if you want to travel during a holiday season. Unlike air travel, you will not get the option to reserve a specific seat. Booking early may also save you the cost of last-minute fares.
On those few trains that don't allow passengers to reserve seats, your best bet is to arrive early. Memorize your itinerary and the times of departure so that you can take advantage of the first-come, first-served policy. Keep in mind that you likely won't be the only one who uses this tactic and that trains may run late.
Most trains will have the option to upgrade your seat -- even after you've boarded -- from coach to either business class, first class or a sleeper car, all of which guarantee reserved seats. Many of these upgrades offer special amenities, such as 'quiet cars' with minimal noise, extra legroom, or -- in the case of sleeper cars -- private bathrooms and sleeping quarters. These upgrades can be quite expensive.
Attendants on Amtrak trains are there to help, and if you're uncertain about where to sit or if there are any available seats -- ask. Many attendants are happy to direct you to a seat, and may group you with other travelers headed to the same destination. Train attendants typically direct passengers away from overcrowded cars in any case, and may assign seats at their discretion throughout the course of a trip.