The AM Track trains going south to Miami and back north stop in Palatka at the old fashion - actually old - train station. I think the train originates in New York City but there are certainly lots of places to pick it up. I do not know if they have any of those tickets that allow you to get off for a day or two before continuing onward, but it would certainly be a neat way to visit our town (To see our Charter School, perhaps) and with a rental car visit other parts of Northern Florida like St Augustine and Gainesville.
Amtrak has a stop in Palatka. The train station is very old and seems nice inside, but it doesn't seem to be open to the public. Warning, call ahead to see if your train is delayed. Apparently Amtrak runs behind schedule very often.
I was amazed when I picked up my friend at this train station. It is an old fashioned kind of train station, actually it is a museum as well as a train station.