Does electrostatic field "go" around obstacles? I want to do electrostatic shielding but I am not sure if blocking only the line of sight is enough.
Imagine that there is an electrically charged sphere, $1 cm$ in diameter, just sitting stationary in free space. Some distance away, for example, $3cm$ is another sphere with neutral charge that I want to shield from the electric field of the charged sphere. 
Is it enough if I put electrostatic shielding: a circular copper plate with $1cm$ diameter between them? Or will the electric field go around the copper that's only blocking the direct line of sight and affect the neutral sphere I want to protect from electric field?
 A: No (but yes).
Your copper shield will have opposite charges on opposite sides as shown here.

The field does not go around the conductor, nor does it go through it.
A: To the question in the title: No, electric field does not go around obstacles. 
What really happens is that the matter (inside the obstacles) is influenced by the field going through it displacing the charges in the matter. 
So, that for example, charges in a conductor displace to such an extent (they can because the electrons are free) that the net electric field is zero inside the conductor. Note that the original electric field does go through the conducting obstacle; it is just that the new electric field (of the conductor) exactly cancels it.
If the obstacle was a dielectric, the charges cannot displace so much that the electric field is exactly canceled. But they do reduce it by a ratio (the dielectric ratio).
To the next part:
A plate alone will not do. The electric field still passes through (as explained above) - it is just neutralized inside the plate affecting the surroundings very insignificantly.
What you can do is cover the complete neutral sphere with a conductor.
