Yes, unlike flights or buses in the U.S., long-distance Amtrak travel includes the expectation that passengers will give gratuities to certain members of the service crew. It is not required (i.e. Amtrak cannot pursue you legally if you choose not to pay tips, and Amtrak staff are not supposed to solicit them), but it is expected.
It is not directly analogous to the situation with restaurant staff, as most Amtrak workers are adequately compensated. It is a longstanding tradition and part of the culture of long-distance rail travel in the U.S., as there was a time when the attendants (largely African-American) did not receive any compensation except for tips.
At the station
If you have a Red Cap assist you with your baggage, then $2 per bag or so is reasonable, same as you would a hotel bellhop.
And that is about it; you do not tip ticket agents or gate agents, for example.
Aboard the train
On board, you do not tip the conductor or the crew operating the train. Even in coach, however, there will be a car attendant who cleans up and keeps coffee and tea available, and so if you are on board the whole day, most people will offer the attendant at least a few dollars.
Celebrity Amtrak enthusiast Jim Loomis, author of All Aboard! The Complete North American Train Travel Guide, has a series of 2013 blog posts devoted to the question for the more involved services. He recommends the following for sleeping car attendants:
For the attendant in a sleeping car, I start with a very basic $5.00 per person per night and, depending on what kind of service and attitude I get, the amount goes up or down from there. So, if I’m traveling with my wife and we’re aboard the California Zephyr all the way from Chicago to Emeryville, that’s a minimum of $20 … two people for two nights. If we ask for or need any extras during the trip – having a meal brought to our room instead of going to the dining car, for instance – I’ll increase the amount appropriately, usually in $5.00 increments. Conversely, I’ll deduct from my basic minimum when a car attendant does less than what’s reasonably expected.
I have seen the $5 baseline figure bandied about since the 1990s, and would bump it up to $10 at this point. It is customary to offer this at your final destination; however, different people have different philosophies. Some tip up front in the hopes of securing a favorable impression; still others do it on a daily basis. I have only done so at the end of the trip.
Regarding dining car attendants:
[T]ip the dining car crew the same way you would tip in any restaurant: start with 15% of the menu prices and go up or down from there depending on the quality of the service.
One other thing: some dining car crews pool their tips and others do not. I‘ve developed the habit of asking at my first dining car meal on any Amtrak trip. If the crew pools its tips – and the best crews usually do – I will sometimes give the LSA (Lead Service Attendant) $20-$30 at the conclusion of my final meal on that trip. Otherwise, I’ll tip after each meal according to the quality of the service.
I will say that in my personal experience, I just see a few dollars left per table at the end of the meal. I don't quite understand why, considering it is table service as you would receive at a restaurant, made more difficult by the tight quarters and the motion of the train. So, I leave 15-20%, but your tablemates may not. Even then, this will rarely amount to more than a couple dollars for breakfast, perhaps $5 for dinner.
As for the lounge car attendant (note: the lounge car is not the same thing as the café car):
[T]he best way to do it is to leave some extra change or a dollar bill whenever you make a purchase, an amount that bears some relationship to the amount of the sale. And, of course, much depends on how you’re treated. Prompt cheerful service is the norm, but every so often, if you get a grouch … well, you know.