Train from Frankfurt to St. Goar

You will always handle your own luggage, carrying it on and off trains (which may involve stairs) and storing it near the end of the train car, on the overhead rack, or sometime in other spaces between seats. This is a good reason to pack light, with each person easily able to handle his or her own bag. See also:

Views inside trains in Part 1 of Rick's Travel Skills show: http://www.ricksteves.com/watch-read-listen/video/tv-show/european-travel-skills-part-1

http://www.ricksteves.com/travel-tips/transportation/trains/on-the-train

http://www.ricksteves.com/travel-tips/transportation/trains/getting-on-the-right-train

A common question that reveals you have never been on a train.
For full details and photos of luggage on trains see here: http://seat61.com/luggage-on-european-trains.htm
Stations and trains do not work like airports and aircraft. Stations are public space, you, or anybody else can walk in off the street and onto a train with nobody to stop them. No check-in, no security, no baggage handling, no nothing. Your ticket is checked on the train after it has left the station.
Conversely, it is your responsibility to get yourself on the right train with your luggage. There will be departure boards and signs to help you (and announcements in German), but nobody will stop you getting on the wrong train. You need to read the signs.
All luggage is hand luggage. When you change trains, it is your responsibility to get off the train at the correct station, with your luggage, and get on your next train.

A train operates more like a timetabled lift (elevator) than a plane.

I was gonna say--the only thing underneath is the track. While you are responsible for all your bags and switching trains at Oberwesel, it is ok to ask and confirm on the Oberwesel platform if the approaching train is your train and is stopping at St. Goar. Also note that the train will likely have the name of its end of line/major destination on it (e.g., Koblenz) rather than interim smaller stops. It's a beautiful route. Enjoy the trip!

"I was gonna say--the only thing underneath is the track."
Trains are not all the same...
From FRA to St. Goar, you might have a transfer in Oberwesel, or a transfer in Mainz, depending on the time you leave FRA.
Trains vary a lot. Trains stopping in St. Goar are local Mittelrheinbahn (MRB) trains that in fact DO have seats like THESE where you can store a bag beneath your seat. As you can see in THIS PHOTO of the MRB, they also have multi-purpose cars for bikes and oversized luggage. These cars typically have flip-down seats LIKE THESE so you can stay with your luggage.

The Mittelrheinbahn connects Mainz to St. Goar directly.

If you ride the Regional Express train from Frankfurt to Oberwesel and change there for St. Goar, you will be changing to the MRB train in Oberwesel. The RE train will likely be a DOUBLE-DECKER (photo taken in Oberwesel.) These trains are tight on luggage space - the overhead racks are inadequate for large bags - but you will probably still make do one way or another.

Russ,
My comment (which granted, I didn't have to write) meant that there's not storage physically UNDER the train itself as a motorcoach would have. I wasn't referring to storage inside the train car. The pics do help with visuals of what trains on that route may look like--thanks for sharing.