what train do I take and how do I take it from Frankfurt Airport, Germany to Baumholder, Germany?

Answer 1

Go to the German Rail website (below). You'll catch the train from Frankfurt Flughafen to Mainz, and then a bus from there to Baumholder.

The train number will depend on your departure date/time.

Answer 2

Best go to the Deutsche Bahn Service Point at Frankfurt Airport (FRA). IIRC they are on level 1 in Terminal 1. You pass them when you go to the train tracks.

http://www.airportcity-frankfurt.de/cms/default/rubrik/13/13146.uebersichtsplan.htm

The staff there will be able to print out the timetable and directions for you as well sell you the tickets for the train. The people there speak very good English and will be able to explain everything. This method is practically idiot-proof.

Alternatively you can use the Deutsche Bahn site at http://www.bahn.de/international/view/en/index.shtml to figure it out yourself. Most connection seem to require you to go by train to Idar-Oberstein (there are various ways to do that) and then to catch the bus either from there or from some other city. Price including the bus tickets should be around 25 EUR one-way. Have that ready in EUR cash. The ticket counter at the airport accepts credit cards, but the conductor on the train and the driver on the bus will take only cash.

BTW, note that there are two airports: The big one aka Frankfurt International Airport aka Fraport (FRA) and the little Ryanair airport called Frankfurt-Hahn (HHN) which 60 miles away in the middle of nowhere and takes an hour to get to by special shuttle bus from Frankfurt main train station.
http://www.hahn-airportshuttle.de/
If you land at Hahn, best take bus to Idar-Oberstein and go from there to Baumholder. http://www.hahn-airport.de/