Train from Frankfurt to Bacharach to Trier

Yes, trains run frequently. Look at the schedule at www.bahn.com.
Frankfurt to Bacharach is a local network ticket. Price is not shown on www.bahn.com, but the schedule is. Prices are at www.rmv.de. Normally you just buy a ticket from a vending machine.
Bacharach to Trier is all within the Rheinland-Pfalz "Land". You can travel with a Rheinland-Pfalz ticket. https://fahrkarten.bahn.de/privatkunde/pauschalpreisbuchen/pauschalpreis_start.post?angebotsgruppe=10&returnurl=https://www.bahn.com/en/view/offers/regional/regional-day-tickets.shtml&scope=standalone&lang=en&country=GBR&dbkanal_007=L04_S02_D002_KIN0060_REGIO-LAENDERTICKETS-KARTE-RHEINLANDPF_LZ03#stay

Note, when looking on the DB website (or anywhere else), the starting point you want is Frankfurt(M) Flughafen
That is Frankfurt (with a u) am Main, to distinguish it from the other Frankfurt in Germany: Frankfurt an der Oder.
Frankfurt(M) Flughafen (the station at the airport) is in two halves, "Regionalbahnhof" (local services) and "Fernbahnhof" (long distance trains. The DB website will indicate which, you just need to follow signs at the airport.

With that info, you can now look up your trains on the DB website: https://www.bahn.com/en/view/index.shtml

Frequency: Trains from Frankfurt(M) Flughafen Regionalbf to Bacharach run every 1-2 hours.

Be aware that Americans have had mixed results with using credit cards at unattended DB stations. So have small denomination cash available if you go to an unattended station. Bacharach was certainly unattended when we were there some years ago.

You might wish to search (top center blue box) this forum for "Middle Rhine" for other ideas and suggestions. You can filter your results (after they appear) for either this forum or our host's website.

Edit: Because of your later post with the actual schedule, I would note that the paragraph just above was meant to include the possibility of a boat ride from Bingen to Bacharach, if desired. We did it without luggage, but many people take their bags. Discount with same day train ticket.

I am not aware of Frankfort in Germany but there are (at least two) Frankfurts as it was already noted here. One Frankfort is the capital of Kentucky. I am aware of another one in Illinois. When you are looking for schedule on Deutsche Bahn website it is necessary to write the correct Frankfurt.

Welcome to the forum wsuec42.

Do you plan to make these two trips on the same day or on different days? That choice will affect your ticketing strategy. So will the number of passengers and their ages, the date, and the time of day you expect to make the trip. Share that information with us if you want some tips.

"What train?" - It's normally "trains" because usually you have a change of train between FRA and Bacharach. You also have a change of train between Bacharach and Trier in Koblenz. When you use the DB itinerary search page, it figures out for you which trains are best and tells you what kind of train each one is as well as what time each one leaves and arrives. So when the search results come up, click on "show details" at the left side of the itinerary box that interests you to see that information. The trains in question will normally be S, RE, or RB trains - the local and regional trains. You might see some ICE or IC trains too - these are high speed trains that are NOT usable with RMV tickets or Rheinland-Pfalz ticket and should be avoided.

Russ
We will travel from a Frankfurt to Bacharach on a Friday and to Trier the next day.

Assuming you are only two adults here... The FRA - Bacharach RMV fare is €12.35 each. This fare is valid at any hour of the day on any day.

No idea what hour you might be traveling... but if you wish to cruise Bingen - Bacharach on your way to Bacharach - and if you can arrange to arrive in Bingen prior to a cruise departure (9:30, 10:30, 11:30, 14:30, 16:30 most of the season) - the RMV train fare to Bingen is €8.60 each. Your train ticket gives you a 20% discount off the normal €15.40 KD cruise fare. The boat gives you the best view of the valley, castles, etc. (I would actually suggest cruising beyond Bacharach to St. Goar for a few Euros more to see ALL of the best part by boat - and I'd suggest overnighting in St. Goar as well, where, unlike Bacharach hotels, some hotels have amazing river views; or after your cruise you can catch a train from St. Goar to backtrack to Bacharach if you definitely want to stay there instead.)

On Saturday you can use the Rheinland-Pfalz ticket at any hour from Bacharach or St. Goar to Trier.

Indeed it's good to have a variety of bills and coins when you buy tickets at the station ticket machines.

There is a ticket office at the airport train station area where you can buy your tickets from an English-speaking human who can explain in detail your available routings. The computer figures it out for you, and the best train route is the one with the most convenient time for you based on when you get to the ticket office. You can familiarize yourself with the daily schedule and see how often the trains run in that direction, but you dont need to tie yourself down until you get there.

Note there is not a manned ticket office in Bacharach so that trip you could book in advance at the Frankfurt Flughafen rail station while you're there. Or buy from a machine in Bacharach.

As previously mentioned, Frankfurt airport has two rail stations - a Regionalbahnhof for local trains and a Fernbahnhof for long distance trains. You'll likely be using the Regionalbahnhof. This website has a map which shows the layout - https://rail.cc/en/frankfurt-am-main/frankfurt-airport-railway-station/l274 .

As I recall, it's not a very long trip to Bacharach. If you can connect with one of the direct (no changes) trains, the trip is only 57 minutes. You can check all the solutions on the Bahn.de website to see which trains will be departing at about the time your flight arrives. You will of course need to allow time for deplaning, walking, immigration & Schengen stamp, buying rail tickets, etc.

The Mosel Valley (Koblenz to Trier) is one of the top scenic sights in Germany & best seen by car because the train often goes out of the valley.

Are you on the RS tour starting in Trier? If so, you will be visiting the Mosel valley on the tour anyway.