I know that you can fly direct to London or Amsterdam from nearby
Austin. Ideally I'd like to go to Europe for a week but I'm open to
other suggestions.
Not much time.
The plan below can be done without any advance booking of trains / tours. Book roos and play the rest by ear. Koblenz (base town) is a rail hub at the nexus of Germany's Rhine and Mosel Valleys, not far from Frankfurt. Think medieval Europe, great scenery, wine. Numbered days below are in no particular order.
Austin nonstop > Frankfurt on Lufthansa July 21-28 (or 30?) Arrive 9:20
Stay 5 nights in Koblenz near the station July 22-27 (Ibis Hotel or apartments for 4 available on booking.com within walking distance) Arrive by direct train from the FRA regional train station (11:24-12:54)
1) See Koblenz' old town zone on foot; hop on the cable car ride to Ehrenbreitstein Fortress.
https://www.visit-koblenz.de/en/public-transit/combo-ticket
(Resource page for exploring Koblenz: https://www.visit-koblenz.de/en/city-tours )
2) Day trip into the Rhine Gorge to Rüdesheim (direct train) - return by river K-D cruise boat (4:15 - 20:00) past dozens of towns/vineyards/castles
3) Day trip to Cologne (direct train)
4) Take a hike: the Rhine Castle trail (west bank of the Rhine) between Oberwesel and St Goar
or... Catch a direct train up the scenic Mosel River Valley to Cochem for Reichsburg Castle + a riverfont bike ride.
5) Tour Marksburg Castle in old-world Braubach (about 12 minutes by train from Koblenz)
6) Direct train to Mainz, near FRA airport for a final day/night.
7) Direct train (25 mins.) to FRA for morning flight. Trains from Mainz start as early as 4 am. But you don't have to leave that early for the 10 am flight I suggested.