Getting to Bingen am Rhein is extremely easy.
Bingen is located about 80 km west to Frankfurt am Main.
Bingen is connected through the German motorways A60 and A61, which are connected to all major German motorways (A1, A3, A4, A5, A6).
It is located on one of Germany's busiest rail road lines.
And finally it's just a quick train ride from Frankfurt Airport. In addition it's possible to fly in into "Frankfurt"-Hahn, served by cheapo-airline Ryanair. If you book well ahead, you may be able to catch a flight for just a couple of bucks from quite a bunch of European cities.
Point your navigation system to Bingen am Rhein. Detailed information on how and where to park your car will be provided later on.
If you plan to go with a major airline, you will arrive at Frankfurt Airport. This is the real Frankfurt Airport, not to be mixed up with "Frankfurt"-Hahn, which is located 120 km away.
Arriving at the Airport, you need to go the regional train station ("Regionalbahnhof"), not the long distance train station. Sadly the way from the airport to Bingen will make you cross the border of two Integrated Public Transport Systems, so most train ticket machines will simply explode if you try to buy a ticket to Bingen from them. They will also burn to the ground if you try to feed some with a credit card, so better bring some Euros with you.
When buying a ticket, you will be asked if you want to go "via Bad Kreuznach". You don't want that. It's a detour and more expensive.
There are two different trains that you can use. All of them are the same ticket and price. Tickets are not bound to a specific train.
RE 2 - SÜWEX: The RE 2 (Regional Express Line 2) train heading Koblenz Hbf is the fastest direct train to Bingen. It typically runs hourly, and is the recommended option. Enter it, and exit and Bingen(Rhein) Hbf, done.
RE 3 - VLEXX: The RE 3 (Regional Express Line 3) train is going to Saarbrücken Hbf, and is a shitty blue Diesel train. Take it, then exit at Ingelheim. Here you need to change trains to RB 26 (Mittelrheinbahn), a grey-yellow train. If you are getting your instructions from the ticket machine it will tell you to already change at Mainz Hbf. Don't do that, in Mainz you would have to cross multiple tracks and carry your shit, while in Ingelheim both train lines stop on the same platform. Exit at Bingen(Rhein) Hbf. Please note this train also stops at Bingen(Rhein) Stadt (City station), which is not where you would want to exit.
Go to the website of Deutsche Bahn to book your ticket. There are InterCity trains stopping at Bingen every two hours. When coming from North-West (Cologne, Netherlands, Belgium etc), you can catch the direct InterCity train from Cologne on. Every other hour, the InterCity does not stop. In this case you will have to change to a Regional Express train in Koblenz, to the RE 2 SÜWEX heading "Frankfurt(Main) HBF". See above for a picture. The IC trains are old and typically are either late or break down, so do not expect to catch your connection in Koblenz. Instead you can resort to RB 26 (Mittelrheinbahn) heading Mainz Hbf. For those regional trains, your ticket is non-binding, even if it was bound to a train for the long-distance part.
If you are coming from another direction with a long-distance train, unless you are connected with an InterCity stopping in Bingen directly, you will typically either change at "Franfurt(Main) Hbf" (the central station), or at "Frankfurt(Main) Flughafen Fernbahnhof" (Airport long-distance station).
If arriving at "Franfurt(Main) Hbf" you can either take the RE 2 (SÜWEX) train heading Koblenz Hbf, which has a stop in Bingen, or take the RE 3 (VLEXX) heading "Saarbrücken Hbf". In the latter case you will have to change trains to RB 26 (Mittelrheinbahn), a grey-yellow train, in Ingelheim. Please note to exit at "Bingen(Rhein) Hbf" (the central station, "Hauptbahnhof"), not "Bingen(Rhein) Stadt" (city station).
If arriving in "Frankfurt(Main) Flughafen Fernbahnhof" (Airport long-distance station) please note that the trains mentioned above depart from the Regional train station, which is a 10 minutes walk away.