East peak in the background
Most climbers arrive in the area in Min Vody and have transport arranged from there to Mount Elbrus. If you travel individually, there are plenty of buses to either direct to Elbrus Village or Terskol or via Baksan. From any of these places you have to continue to Azau (2335m), the end of the road and the start of the climb.
Most people travel the first part of the peak in the cable cars, but some walk all the way on the paths of loose scree. The last part of up to "Barrels Hut" can be travelled in a chair lift. You are now on roughly 3900m and this is where the climb starts for real.
Monuments on the east peak's flat summit plateau.
Going to Russia only to bag one peak along a very well travelled route on one of the world's most popular mountains?
Why not traverse the peak and climb both the summits?
Some old artifacts from the Soviet era is to be found on the east peak and you get away from the hordes for an hour or two. It's also a good spot for shooting some good photos of the main summit.
From Barrels Hut to Priyut 11(a.k.a. Diesel Hut) 3900m - 4150m
Very straight forward walking on hard packed snow. Watch out for skiers bombing down at high speed on this section. The last part to Priyut 11 is the steepest. You easily walk this stage under en hour.
Priyut 11 to Pashtukova Rocks 4150m - 4650 to 4800m
Walk in between two band of rocks leading up the mountain's gentle slopes. On the right hand side you can find good tent platforms amongst the rocks. You can also pitch you tent after a steep section en route, where you find some larger areas of rocks. This place marks the end of Pashtukova Rocks.
Pashtukova Rocks to the east peak of Elbrus 4800 - 5621m
Climbers on the west peak, seen from the east peak
Start the traverse towards the left along the normal route to the west/main peak. When you see a large snowfield above, start making progress towards it and leave the normal route. If you find it too steep (some very short sections of maybe 55 degrees) you can always switchback your way up. A ridge will appear above you and when it gets too step to go on towards it, head right. Look for and easy passage through a rocky part. As far as I could see, there are plenty of alternative options. When you have passed the rocky section it gets less steep. Still walking right on the snow, start looking for the path of least resistence towards the summit. There are two points that appears to be of equal height. Aim for the one with some Soviet era monuments on. The last part is a very easy walk where you hardly gain any altitude.
This one I didn't climb myself, but it sounded simple enough, when my friend explained it for me. Instead of traversing left towards the west/main peak, you start traversing right. Follow the mountainside slightly upwards until you find the gradiant towards the summit comfortable. Big snowfields will take you to the summit.
Climb the normal route to the saddle. Study the mountain side a bit before taking off. There are some rocky steep sections where you don't want to end up. The route I climbed is located on the left hand side of the "face". Easy walking on snow most of the way to the summit.
All of these routes are also suitable for descent.
When I say: "too steep" in the sections above, I talk about too steep for climbing without rope.
Warm clothing, walking sticks/one long ice ax, crampons, sunglasses, water&snacks and compass/GPS.
Two things to think about
1) If you attempt this route you're strongly recommended to notify the rescue personal or the administration in Barrels Hut about your intentions. There have been climbers supposedly missing on Elbrus, who were delayed on the eastern summit, were fine, but the alarm about missing climbers went off because no one knew they were there.
2) When heading down from the peak the normal route, be sure not to walk to far down. Stick to the traverse until you're sure Pashtukova rocks are below you. If you head down too early/too direct from the saddle, there is a risk you walk into a crevasse or fall on some steep sections.