Question

How does a free electron at rest move in an electric field?

Answer 1

As the electrons are free to move they do so until they find positions where they feel no net force. When they come to rest the interior of the conductor must have zero electric fields. This means
1) The macroscopic charge density inside a conductor is zero.
2) The net charge on a conductor only exists on the surface. (at least to good approximation as the electric field will penetrate slightly into the conductor).
3) External electrostatic fields are always perpendicular to the conductors surface. Otherwise this would produce a force on the charge carriers inside the conductor and so the field would not be static as we assume.

4) The electrostatic field at the conductor's surface is proportional to the surface charge, i.e. does not depend on the charge carriers inside the conductor. 

Answer 2

In an electric field an electron always moves in direction opposite to that of an electric field.