It depends on which train you take, but if you go Southwest Chief to chicago, Capitol Limited to Washington and then Silver Meteor to Ft. Lauderdale it would be 2 nights on Southwest Chief, 1 night on Capitol Limited and 1 night on Silver Meteor.
The online reservations system is going through maintenance, from time to time they have to do this to add new updates, etc. It should be back later today or tomorrow morning. You can always call the 800 number to make Reservations.
I hope this answered your question. Also, if it seems expensive to you, check out the 15-day rail pass which includes 8 segments.