Nearest Amtrak rail station is in Kissimmee (next door to the Greyhound station also, but of course the train will be much, much more comfortable). The Silver Meteor. train 97, departs Kissimme southbound daily at 1:32 p.m., arriving Delray Beach at 5:23. Train 91, the Silver Star also makes the trip, but takes a side-trip to Tampa in the process, making it a longer journey.
You can also catch the train in downtown Orlando, on Sligh Street, but Kissimmee is closer and more convenient.