"We are arriving approx 7:45, and was hoping to catch one of the trains between 10 and 10:30."
I want to get into the details here so you can make an informed choice. Let's review what the DB website offers...
Trains leaving FRA between 10 and 10:30 for Rothenburg... ARE THESE the ones you are considering?
- 10:01-12:51 using the ICE train from Frankfurt Airport's "Fernbahnhof" station (2 transfers to reach Rothenburg)
- 10:09 - 12:51 using the ICE train from Frankfurt Airport's "Fernbahnhof" station (3 transfers)
- 10:34-13:51 using the ICE train from Frankfurt Airport's "Fernbahnhof" station (2 transfers)
If so... Although the QdL ticket which Sam suggests is inexpensive and totally flexible, it does NOT permit you to use the faster ICE trains journeys; it covers only regional train equipment. The full fare on these ICE trips will be about €72 each (assuming DD is adult) if you buy them at the airport train station. To find the "regional-only" journeys for the QdL ticket you must do this when you search...
- click on "mode of transport"
- select "local transport only" from the pull-down that appears.
Sadly, only one journey in your time frame will appear - 10:09 - 13:51, with "BUS X61" at the start of your trip (with 3 transfers.) The starting point for this bus is obviously not a train platform inside the station. It's a bus stop titled "Flughafen Terminal 1, Frankfurt a.M." The bus takes on a 21-minute ride to one of Frankfurt's many stations- the "Südbahnhof" - where you switch to a train.
The NEXT regional journey option is 11:09 - 14:51, with the "HLB" RE 59 title. This in fact is a TRAIN that leaves from the airport's "Regionalbahnhof" station. (This is the airport's other train station - the one for regional train departures.)
So... Here's how I suggest "working around" DB's website to find schedules for the QdL ticket... Break it into two search segments with Frankfurt's central station in the middle.
1) Set mode of transport to "local transport only."
2) search FRA airport > Frankfurt (Main) Hbf (Frankfurt's central station.)
You will see there are 12-13 different direct train journeys between 9 am and 11 am. Each takes about 13 minutes. All leave from the Regionalbahnhof. After you have your QdL ticket in hand, I would just take any one of those departures whenever you are ready to leave the airport.
3) search Frankfurt (Main) Hbf - Rothenburg od Tauber
You will see these 3 regional train journeys each will have two changes of train - one in Würzburg and one in Steinach...
9:30-12:51,
10:34 - 13:51,
11:30-14:51
Take whichever one of these is the next departure from the central station.
My hunch: you'll probably get there at 13:51. But you might arrive earlier if all goes smoothly. And you won't have to use the bus - at least not at the BEGINNING or your train journey.