Elbrus and Cheget
The main settlement is Terskol, which is at the bottom of Elbrus mountain. From there you can walk (or drive) to Azau station (about 4 km), which is where the lifts to Elbrus are. In about 4 km from Terskol there is Cheget tourbase, where there are lifts to Cheget mountain. Terskol is a small southern settlement and it's very hospitable: there's lots of accommodation, cafes, and services for all needs. In terms of nightlife it might get a bit rowdy, so it's better to avoid that bit. The local population is mainly Balkarians, friendly and temperamental people. Even though it's quite a calm place, some people go to Elbrus only in a group of people, but it has more to do with the risk of getting in avalanche. If you are wondering that Chechnya is not too far, forget about it: there's a mountain range in between, it's more than 500 km away, and there has never been a problem anyway even during the war.
There are a lot of cafes and restaurants in the main village Terskol, as well as Azau and Cheget tourbases. There are also some stands both on Elbrus and Cheget mountains, don't be put off by their run-down looks: the food in most of them is great and cheap.
If you want to live as close to Elbrus lifts as possible try Azau hotel (tel +7 928 2796212 or +7 928 2575384) or Krugozor hotel (same tel), which is even higher. Both are wooden cottages with all the amenities and are actually quite comfortable. The price is 500 R ($17) per person. There's another chalet called "Logovo" at Azau plain (tel +7 866 38 7 11 12). If you want to live as close as possible to Cheget lifts, then Cheget tourbase (+7 866 38 71339) is your best bet. Even though it looks like a purpose-built block it has quite comfortable accommodation if you're prepared to pay (from $35 for a room) and there's lots of services inside. Among other options are the luxurious Balkaria hotel, Terskol hotel (in Terskol settlment, tel +7-866-38-7-11-40, +7-866-38-7-13-22), there are also many other smaller hotels and you can find a list at http://www.elx.ru/residence.htm If you already tried calling the numbers above you found out that (surprise!) the people there don't speak English. So, unless you have basic Russian skills, you can try to book your accommodation through the local travel agency http://www.go-elbrus.com (run by a German - quality and price) or email@example.com (which is not really an agency, but might help).