At the major terminal stations, you will often be directed to a gate number rather than a platform, and an agent at the gate will check your ticket/ID, and issue you with a slip of paper, the colour of which will indicate which group of coaches you should go to (just ask an attendant if you are unsure).
The seating within the coaches is determined by the attendant for that coach.
Almost all Amtrak tickets are now issued as e-tickets, so the gate agent or train guard will just scan the QR code on your ticket to confirm you are aboard.
There can be quite a scrum for getting through the gate (much like getting aboard an airliner), so heed the advice to be at the gate 30 minutes before departure.
At the smaller stations, it is not uncommon for only one or two doors to be unlocked, the guard or attendant will most likely give your ticket a cursory examination, to be sure you are on the right train, and direct you where to sit. The guard will come around later and scan your ticket.
A few other pointers:
The only alcohol allowed aboard is in the café/restaurant, or a private cabin if you have one.
Because of the length of journeys, the long distance trains stop every few hours for a smoking/leg stretch stop, if the timetable allows.
The TSA occasionally search bags prior to boarding, although I've never seen it. I believe Amtrak policy is that if you object to any search, you can have a refund (Amtrak and the TSA have an uneasy relationship, and Amtrak have been known to throw TSA agents out of their stations for not following railway custom)
If you're on a long distance train with checked luggage, providing both your arrival and departure stations have the appropriate facilities, this comes highly recommended (and is free). If you have a morning train, they are often willing to allow you to check your bags the night before, which is handy if you are traversing a large city in the morning rush hour to get to your departure.
Remember that Coach = Standard (class)
If you're travelling in the western US, the Superliner coaches are very comfortable, even if the track could do with a bit more maintenance to give a smooth ride. The observation car is definitely recommended. There's no such thing as first class seating on these trains, quite frankly the reclining seats on Superliner coaches are as comfy as many first class seats in Europe.
If you're on a budget, Amtrak cafés are not cheap. It's not unheard of for passengers to bring along a coolbox with a few meals' worth of food on ice. On the multi-day trips, some of the crew-change stops are lengthy (can be an hour or so) -- the guard will announce if it's ok to wander into town to find a more substantial discount repast -- although you should note that some US downtown areas are less salubrious than European ones, and mini-supermarkets are much less common in downtown US.
However, if you have a few $ to spare, Amtrak is one of the few western railway systems where you can still get a full three course meal served in a proper restaurant car on all very-long-distance trains. The maître-d' will come through the train a few hours before each meal to take reservations.
If you're meeting a train midway through its journey, you can sign up for email alerts through amtrak.com to confirm that it's running to time (smaller towns often just have a platform with no facilities - not somewhere you want to be standing in the rain for four hours awaiting a delayed service.)
In general, you will find that while Amtrak is somewhat inefficient, the staff in particular are very friendly and welcoming, and Amtrak policies & practices are generally quite customer-friendly. Enjoy the scenery, and don't fret if your train is an hour or two late!