EM Waves in Coaxial Cable
Yes, we normally assume electric field goes to zero inside a good conductor (otherwise very large currents would be produced). As Eugene says in comments, this provides a boundary condition when calculating the fields in the dielectric.

In a non-ideal conductor, the electric field does penetrate somewhat. The depth that it penetrates is called the skin depth, which depends on the signal frequency and the conductivity of the conductor. Calculating skin depth is important when determining the loss of the coaxial cable and how it varies with frequency.