Both winter and summer, go sightseeing in the Elbrus: take the gondolas / cabins to the top 2nd station. Do not take the chairlift but climb on one of the snow-cats waiting for tourists. The snow-cat will take you right to the Elbrus base camp at 4200m. The view from there is awesome. Of course, check the price prior to climbing on the snow-cat, if you find it abusive, you always can take the chairlift. it may be possible to convince some of the snow-cats operators to bring you to the top of the glacier at 4600m.
Many hiking possibility in Elbrus vallley and Cheget but also much closer to Nalchik 1h drive in Bezengi valley and Chegem valley. The mountains there are not as high more pre-alps type but the sceneries and the flora are just as beautiful and you will not a meet any tourist at least after you pass the chegem waterfalls. A recommendation would be Bezengi.
In any case, don't think you will get nicely marked pathes like in the alps, the sierra nevada or the rockies. Just get yourself a good map in a bookstor the best found is a 1:200'000, choose which valley you feel like driving to, in this valley identify a path on the map and go for it. Alternatively simply drive to the valley and choose a path going sideways. The mountains are used for livestock rearing in the summer so that there are always plenty of shepard and cow-boys. You can't really get lost. You will find tracks made by the animals and pathes. These may not always lead you to the summits but will get you to the passes the locals do not climb the summits but do need to pass from one valley to the next with their stocks.
In winter you can do snowboot hiking in any of the a/m places. However it is safer to stick to the Cheget / Elbrus area which is more populated in winter.
