Is the direction of an electric field opposite to the direction of a current?
I consider electricity to be an observation of the property of electron mobility.


This isn't a good way to state it.

If you get into solid state physics, you'll learn that the term electron mobility is already used to mean something else. It's the property of a material that determines how readily its free electrons can move. It's one of the factors (along with the density of free carriers and the charge of the carriers) that determines the conductivity of the material.

You should also consider that electrical phenomena include electric fields generated by magnetic fields rather than by charge, propagating electromagnetic waves, etc.


  If electrons go the opposite way of the electric field, what is the connection between the electric field and an electric current in a simple planar circuit?


This is given by the microscopic form of Ohm's Law:

$$\vec{J}=\sigma\vec{E}$$

where \$\vec{J}\$ is the current density and \$\sigma\$ is the conductivity of the material. 

As mentioned above, the conductivity (for a metal or n-type semiconductor) depends on the density of electrons, their mobility in the material, and their charge

$$\sigma = n_e \mu_e e$$

where \$n_e\$ is the electron density, \$\mu_e\$ is the electron mobility, and \$e\$ is the fundamental charge (the charge on each electron).

If you consider p-type semiconductor or ionic conductors you will have to consider conduction due to positively charged particles, not just electrons.