Does the electric field travel mostly on the surface of the copper in a standard circuit?
I read somewhere that the electricity flows mostly on the copper surface.


When high frequency AC signals are present, the current in a wire flows mostly near the surface. This is called the skin effect. It's main effect is to increase the apparent resistance of the wire as the signal frequency increases.


  Are they talking about the electric field or the drift that is mostly on the surface?


Probably currents, and only at high frequencies. Significant electric fields tend to be found mostly outside the copper altogether. 


  I assume since the larger the wire the greater the EMF that can be forced through


EMF does not go "through" things. EMF is measured between two points (possibly as an integral along a particular path between two points).


  Since it is not possible to have an electric field with out a magnetic field is a radio wave being sent out each time an electric circuit is turned on and if so WHY DOES IT NOT transmit continuously ?


You are correct that generally when there are AC signals in a circuit, it will generate a radio wave. 

However when we learn circuit theory, to keep things simple we start out by learning about lumped circuits. This means we assume the circuit is physically small compared to the wavelengths of a radio wave at the frequency of the signals in the circuit. A consequence of this (and the reason for making the assumption) is that the circuit will not radiate very efficiently, so that we can neglect the power loss due to the (very small amplitude) radio waves being produced.

In the real world, this assumption may not be valid and we do indeed need to worry about these radiated radio waves, both because they affect the circuit operation, and because they might interfere with other circuits (say, a radio receiver) nearby.

These are behaviors you'd learn about when studying electromagnetics or electromagnetic compatibility.