I'm not sure what you mean by alternatives. There's just Amtrak -- you won't have any other options and there's probably only one a day or even just a few a week. I don't know about discounts. I used to get one when I was a student but that's it.
It's a good thing you're flexible with time because an Amtrak trip of that length could literally be a day or two late. They don't own all of the tracks so sometimes you have to just sit and wait for hours for freight trains to go by. Even when I used to take it between Michigan and Chicago in college what was supposed to be a 4 to 5 hour trip could take twice as long. I hate driving long distances, but Amtrak is so unreliable for big trips like that I'd much rather drive and take my time (like split up the driving over 3 days) rather than take a train.
Travel is fatal to prejudice, bigotry, and narrow-mindedness, and many of our people need it sorely on these accounts. Broad, wholesome, charitable views of men and things cannot be acquired by vegetating in one little corner of the earth all one's lifetime. Mark Twain
Ditto mdgirl on alternatives. There's just Amtrak, as far as I know, in addition to commuter trains that connect areas like Miami and West Palm Beach.
The longest Amtrak trips I've done are Philly to Raleigh going south and Philly to Montreal going north, but I just sat in regular seats the way I do for shorter trips. I don't know if there are sleeper cars, to be honest. I sleep fine on trains, but I also sleep fine in cars and pretty much anything that is in motion, so my requirements aren't very high. If you don't think airline seats are comfy, I am guessing you won't think much of train seats. (Although, like I said, I don't know if there are sleeper cars. I've only done sleeper cars on European trains.)