Train from Frankfurt Airport to St. Goar

Yes, www.bahn.com, the German railway site. Don't know what is "weird" about it. It does show "no special fares available" for most trains, but there are flexpreis tickets (meaning anytime tickets) of 39 EUR for 2.
While in St Goar you can travel on VRM tickets, either individual day tickets or minigroup tickets over the entire network for cheap. Network goes south to Oberwesel, up the Mosel to Bullay, north along the Rhine to close to Bonn, as well as trains on the east bank of the Rhine.
http://www.vrminfo.de/en/timetable/

You can get tickets for 19.50€ p.p on your day of travel, no reservations needed. Using the Regional trains from the Frankfurt airport to St. Goar.

Bahn.com will eventually take you to reiseauskunft.bahn.de/bin/query.exe/en. Why not just start there.

What do you mean by "explains the local trains". There is a lot of information on the query webpage. For a specific connection, click on "details" to show you the trains used in the connection. Then click on the train number to show you it's entire route and times. The page also shows the times between trains and in some cases the track numbers. Sometimes the Bahn website will not show fares because they are part of a "metro district" (Verkehrsverbund). For those fare you need to go to that metro district's website, www.rmv.de for Frankfurt to Bacharach and www.vrm.info for Oberwesel to Koblenz (and beyond).

You seem to have found your information on Google maps or a similar site. Those will give you an idea as to the trains on a certain route, but you can't use them for reliable information, so that's where your "weird results" come from.

The Deutsche Bahn website is the place to go to for reliable information on trains.

"We are spending four nights along the Rhine and will be making some day trips."

A very nice amount of time. Good choice.

Sam gave you a link to the VRM website, which used to be VERY informative, but after recent changes now has next to no information at all in English about tickets, etc. It is likely to disappoint and frustrate you.

But the VRM area covers most of what most people want to see of the Rhine/Mosel region - Cochem, Burg Eltz, Marksburg Castle in Braubach, Rheinfels Castle in St. Goar, Boppard, and Oberwesel, Remagen and Linz. Here's the most recent map (light green area.)
http://www.vrminfo.de/fileadmin/data/pdf/2017/VRM_Schienennetzplan_2017_NEU.pdf

There are 3 railway lines running south out of Koblenz - the one to the southwest is the Mosel River route. The other two are the left and right-bank routes for the Rhine. The lines running NORTH out of Koblenz follow the Rhine on the left and right banks too.

The VRM day tickets (€18.80 each) and mini-group tickets (€22.10 for 2-5) can be used within the light green zone; they are explained somewhat incompletely on the following page:
http://dev76.vrminfo.de/en/tickets-and-fares/ticket-offers/leisure-ticket/

Both are valid on local and regional trains for outings (after 9 am weekdays, any hour on Sat or Sun.)

There's also a 3-day pass for at €37.60 each. It's valid on the same trains but is valid for a full 3 days with NO 9:00 am restriction.

The Rhineland-Palatinate ticket/day pass (€24/1 + €5 per additional passenger) is an other option that covers a much larger zone and would included trips to Trier, Bacharach, Bingen, Rüdesheim, Bonn, Mainz, and many other cities. Map and info. are below.

http://www.vrminfo.de/fileadmin/data/pdf/2017/RLP-Ticket_streckennetz.pdf
https://www.bahn.com/en/view/offers/regional/regional-day-tickets.shtml

Feel free to ask any specific questions.

Don't remember details about the ticketing as such, but when we flew in to Frankfurt we bought tickets to St. Goar at a train ticketing office. Don't remember if it was the S-bahn or not, but we did change in Mainz and rode along the river to St. Goar.