It takes 9 hours to go from Omaha to Miami on Amtrack , and you can get a one way ticket for $275.
Yes, you can opt for the 23 train that takes two partial days (not three full days - although there might be a possible package for that too), and yes you can buy a cabin ticket for almost $1100.
No it's not faster to drive, and I've driven such distances. Cheaper? Perhaps. If you have more than one person, definitely. Again, I've done this, I prefer to drive, and often get a rental with full coverage, in case I decide to pull any Jackass stunts ( with full coverage, I can take the car to a demolition derby before returning and not have any worries). It's definitely not more relaxing, especially if you're trying to beat a train going 90.
Note: if going from Omaha to Miami you'll probably go first to Chicago, and may get put on the City Of New Orleans (made famous by the song), and go to , you guessed, it New Orleans, then to Jacksonville, and then to Orlando, and then to Miami, there might be 3 to 5 train changes there. Then there are other, slower routes, with more changes.
It may be more rewarding to drive.
Also, it should be noted that some train stations have TSA agents and you'll still have to deal twith them sometimes by train. If you go that route. Trains can be fun though, no need to turn off your electronics, and you'll likely have excellent signal strengths wherever possible, plus a lot more room to get up and walk around and socialize.