Two things to check for steering wheel vibration:
1. Check your front wheel bearings for play and re-torque them if needed.
2. Rebalance front tires
To check that upper ball joint remove that "goldish" nut and separate the upper control arm from the knuckle. Once off move the ball joint around and check for clearance. I am guessing that it will be somewhat hard to move (dried up). If there is no clearance you can try re-greasing as I described earlier or you can change it. This is a pretty simple DIY project.
As for lower ball joints they can easily be checked without removing anything. If these need to be changed you better leave it to your mechanic.
I doubt that you have a shocks problem. BTW, how many miles on your car?