Although a well organized system of transportation, Amtrak actually functions differently based on route, train and location. Generally, stops on the West Coast are far more frequent than stops in the Southern or South Eastern US. Long rides in the South West consist of less stops with longer stops in each city. The West Coast Surf-liner features frequent stops and very short pauses between stops.
If you are are boarding the train somewhere between New York and Georgia, it is quite possible that you will only pass 5 to 6 stops per state with a brief pause at each station. This system involves a higher level of attentiveness as arrival times will tend to vary from 20 to 30 minutes on average and there are longer breaks between cities.
On most occasions an attendant is likely to wake you before your stop if you do fall asleep, but it is not always the case. I would recommend an alarm clock or notice on your cell phone to make sure you have time to gather your belongings before departing the train. There are generally 5 to 10 minutes alloted for boarding and departing the train depending on your station and if the train is on time or not.
Some trains have cafe cars or snack stations but the food can be a bit costly at times. One recent development has been the discontinuation of free snacks in favor of vending machines. Currently, these train cars do have free water and coffee while charging for soda and snacks. Of course in these cases credit cards are not going to be much use so I recommend packing some change just in case. There is always the option to bring your own meals on board as well.
While laptops are always a great way to pass the time on the train, be aware that trains do not currently carry wifi. If you forget to bring a dvd for your laptop don't worry: the view outside can be quite fun to look at.
Just like any form of transportation, be aware of your departure and arrival times. Amtrak can be quite relaxing if you arrive an hour before departure time and don't bring too many carry-on and check-in bags.