Elbrus and Cheget
The journey is quite long and it is not advisable to go by car because of the poor road quality. However, if you're up for it, you need to go along the federal route from Moscow to Rostov-na-Donu, then drive to Mineralnye Vody and from there it is a 2-3 hour jouney by car to Terskol.
It's the best option. Both Aeroflot and Sibir Airlines have direct daily flights from Moscow to Mineralnye Vody, which take 2.5 hours and cost about $200 return. You then need to take a 2-3 hour taxi. If you book a tour, you might get a direct flight from Munich to Mineralnye Vody.
Local buses, shuttles and taxis depart from Mineralnye Vody. The journey takes 2-3 hours, costs about 200 R ($7) by bus, about 1000 R ($35) by taxi. Beware of local taxists, as their main entertainment is to bargain, so you'll have to chat for half an hour, before you can drive. Welcome to southern Russia. You need to take a taxi to Terskol (the nearest settlement to Elbrus mountain) or Azau tourbase (4 km from Terskol, just next to the lifts) or Cheget tourbase (that's closest to Cheget mountain, which is about 3 km away from Terskol).
The closest train station is Mineralnye Vody (Stavropol region of Russia). Daily train #34 from Moscow Kazansky station to Vladikavkaz bring you to Mineralnye Vody in about 28 hours. A better train is from Moscow Kursky to Kislovodsk (#27) also takes about 28 hours. The price is about 2200 R ($80) one way in 2nd class.