DAY 1
Yes, use the Regionalbahnhof station at FRA.
Buy one-way RMV tickets from the FRA ticket machines that Sam suggests on Day 1 (if you will travel only as far as Bacharach that day. If you are also going elsewhere, you need different or additional tickets.)
Here's a photo of the ticket machines with the RMV logo.
TRAINS: You can ride ANY of the regional trains between FRA and Bingen with this ticket - S-bahn, RB, RE, MRB. You do not reserve these - you just get on. There's no point in reserving anyway because you can't pinpoint a time after a flight.
DAY 3
- You should buy the RMV group day ticket from a Bacharach ticket machine for the round trip (€34.50.) Only a day ticket allows for round trip travel, no extra charge. It is valid for the two of you to FRA and for the FOUR of you back to Bacharach. Again, if traveling elsewhere the same day, you might need a different ticketing strategy.
Most trains between FRA and Bacharach involve a change of train in Mainz. There are frequent trains. If you want a DIRECT train, then use the RE trains from FRA at 7:21, 9:24, 11:23, 13:23, 15:24... (Before, after and in between there are multiple journeys from FRA where you have a connection.)
The RMV website ticketing and scheduling pages will confuse you, and you don't need it. Use the DB page to find your train schedules:
http://reiseauskunft.bahn.de/bin/query.exe/en
When you search, be sure to check the "only local transport" box to find trains that will accept your RMV tickets.
Frankfurt(M) Flughafen Regionalbahnhof is the name of the station to enter for FRA airport. Just Bacharach for your destination.