"Is it easy to get a train to Bacharach?"
Yes. Direct regional express trains take less than one hour; leave Frankfurt airport's Regionalbahnhof station for Bacharach at 9:24 and 11:24. There's also a 9:29 train, a 9:59, a 10:29, a 10:37, and a 10:59 in between - these will have a layover in Mainz.
Follow directional signs inside the terminal to the underground Regionalbahnhof.
Ticket machines with touch-screen, English option. A RMV group day ticket for €34 gets you all to Bacharach.
Regionalbahnhof platforms. Note luggage carts in case you fail to pack ala Rick Steves' advice.
Mainz' modern station, lots of shops, cafes.
Bacharach's small outdoor station
You can probably get a car in Koblenz at the main railway station if you arrange that with Autoeurope. I dropped a car once at a dealership just behind the station there. Emmelshausen is another Autoeurope option - a small town not far from Boppard (another Rhine village I like enormously) that you can reach by a scenic railway line (steepest tracks north of the alps, they say.) However, you could just as easily travel from Bacharach to Würzburg (near Rothenburg) by train (saver fare ticket) and pick up a car right there at the station. I did that myself about 15 years ago. (Not that anyone really needs a car for the area around Rothenburg either.)
The train normally offers improved sightseeing since it separates you from the asphalt and road traffic - and the one who would otherwise be driving can enjoy it as well. You can picnic as you go. Or even enjoy a glass of wine. No potty stops or relegation to the back seat for companions. Fun for people-watching too. (As a slow-to-learn sort, I used to drive Germany, but I finally graduated.)
BTW, If Maria Laach (which I suspect Rick has never visited) is actually on some train-traveler's list of destinations, a car, while handy, is not an absolute necessity. Catch a train along the Rhine to Andernach; a bus takes you from there.
One place I'd strongly suggest for families (or anyone, really) near Rothenburg is the Franconian Freilandmuseum (Bad Windsheim) with its excellent collection of centuries-old farmhouses and other buildings.