How does the speed of electricity become the same as the speed of light? 
"The field due to the battery sets up a surface charge in the wire. The surface charge is negative near the negative pole of the battery, and positive near the positive terminal, and varies more or less linearly along the wire. The surface charge in turn sets up inside the wire an electric field which is constant across the diameter of the wire, and along the length. This field accelerates electrons"

If there are charges being set up in the wire, the  charges (electrons) have mass and cannot move at $c$. How then do they not affect the speed of electricity and gives us its speed to be equal to $c$?
Do electrons in a wire (load) affect the speed of electricity due to their own velocity?
Think about a circuit with a 5 V battery and a very long resistor with two switches at its end. So, when both switches were open there was no charge setup inside resistors. When we close both switches then, the charge will setup and the signal travel at the speed of $c$ inside that resistor. How?
Doesn't the charge being set up and their own electric fields being created slow down the signal's speed?
 A: The information about the disturbance of static charge equilibrium (toggling the switches) travels at the group velocity of light in matter, which is in turn related to the refractive index in matter, or more precisely to its dispersion (rate of change of refractive index with frequency/wave number). Group velocity is sometimes close to the speed of light, but sometimes significantly smaller (but never greater, which would be a violation of causality/special relativity; for causality in material electrodynamics see also Kramers-Kronig relations).
Models of dielectric permittivity (and hence, refractive index) in matter, namely in conductors, are the Lorentz and Drude models (see this reference for a compound model). These models also show how the mass of electrons enter the dispersion relations. In other words, the fact that the electron mass is non-zero, is encoded in a more or less convoluted form into the dispersion relations (and hence the speed of light) of electromagnetic waves in matter. They determine the slackness of the electrons to wiggling around in the Lorentz oscillators, so to say, which determines the effective speed of light in matter.
Needless to say that even if the speed of information is somewhere around (but, probably well, below) c, the relaxation time to regain static charge equilibrium can be considerably longer than the time to travel the whole circuit. This is because the disturbance causes oscillations, which are damped depending on the properties of the individual circuit. There is resistive damping, and there is radiative damping, all of which determine how long it takes the stirred up charges to calm down.
A: Electrons don't need to bump into each other to transfer movement, like billiard balls. They are charged particles which interact with each other at a large distance through the electromagnetic field.
Imagine a room packed with balloons: if you push on balloons on one side of the room, the wave travels several meters to the other end of the room in a matter of seconds, even if none of the balloons traveled more than a few centimeters. In this analogy, balloons represent electrons together with the field they produce.
This is why the electrical signals travel at e.g. 0.7c, while individual electrons move at less than 1% of that speed (depending on the temperature), even at microscopic level. At macroscopic level, the charge drift happens even slower, with typical speeds in mm/s or cm/s, depending on the charge and current densities.
A: To answer your title question, the speed of electricity does not become the speed of light. But many "messages" or, more correctly, transient effects that set up the final steady state flow do indeed travel at the speed of light in the medium your circuit is steeped in
Think of your circuit at the very instant you close the switch. The switch changes a boundary condition of the electromagnetic field, suddenly (let's say for simplcity, instantaneously) imparting a potential difference between the two conductors at the battery end, whereas the potential differnce just before the switch close was nought.
Let $+z$ be the direction along the conductors. For simplicity we think of your conductors as a waveguide whose cross section is translationally invariant in the $z$ direction.
If you solve this boundary value problem given that the initial Potential Difference (PD) was nought but is now suddenly +V at one end of the conductor, the solution is that the potential difference now travels down the conductors as a freespace electromagnetic wave, progressively raising the PD from 0 to +V at points of increasing $z$ as the wave travels.
The speed of this wave is not simple: the waveguide comprising the conductors with freespace between them have a system of eigenmodes with different speeds that depend on the cross sectional geometry. But the lowest order mode, the TEM mode (see footnote) for a two conductor system, indeed runs at speed $c$ if the conductors are steeped in free space. The disturbance comprises eigenmodes that travel at speeds $c$ and less than $c$.
As this disturbance travels, the EM field acts on the conduction electrons in the wires. They begin to move. They can't move at $c$ as you rightly point out because they have nonzero rest mass. As they accelerate, they generate their own radiation: new electromagnetic waves that travel down the conductors as a system of modes with speeds $c$ and less.
Meanwhile, the freespace wave reaches the resistor. If the resistor is not matched to the waveguide characteristic impedance, a reflected wave begins to come back, comprising eigenmodes at speeds travelling $c$ and less.
And all these waves bouncing back and forth move the electrons which radiate their own reaction fields. We get, very fleetingly, a hugely complicated system of waves bouncing back and forth,coming from both the original closing of the switch as well as radiation from the accelerated electrons.
Eventually these transients settle down with the electrons moving uniformly through the conductor at their drift velocity set by the potential difference $V$. This is MUCH slower than $c$. There are no more bouncing waves: the electromagnetic field is now static and the electrons drift in their boring old constant speed pattern around the circuit.
But, because the initial transient includes the TEM eigenmode that travels at $c$, the first motion of electrons at the resistor end of the system indeed begin to move a time $L/c$ after the switch is closed, where $L$ is the $z$-direction distance along the conductors from the battery to the resistor.
Footnote: TEM ("Transverse Electromagnetic") modes arise in two conductor waveguides, i.e. those with a froward and backward "return" path, i.e. a system that can be thought of as a closed circuit. For conductors in freespace, they always have speed $c$ and, curiously, the cross sectional EM field configuration is the same as an electrostatic / magneto static configuration, but this cross sectional static variation is subject to an aplitude change that moves as a wave, as i explain in  my answer here and also here.
A: 
If there are charges being setup in the wire, the charges(electrons) have mass and cannot move at C. How then they don't affect the speed of electricity and gives us its speed to be equal to C.

They do affect the speed.
The speed of the signal in a cable or other communication channel is given by the velocity factor which is the speed of the signal expressed as a percentage of c. This is 100% for a radio wave in vacuum, but it is about 75% for CAT 7 twisted pair cable. The velocity factor primarily depends on the insulator rather than the conductor. So a coaxial cable with air as the insulator might propagate signals at 93% of c, while one using polyethylene might have a velocity factor of around 80%.
The velocity factor is sometimes considered to be the speed of light in the material as opposed to c which is the speed of light in vacuum. However, it is a little complicated since the structure is not just a simple homogenous material (the dielectric) but a complicated waveguide.
A: The electrons themselves aren't moving that fast, there is an electromagnetic field that makes electrons move in a certain direction within a conductor.  It can be helpful to think of it as a pressure.
It's not equivalent, but think if you hook up a hose and fill it with water.  Then you pump more water into one end.  The pressure will make water come out the other end very quickly, but he water molecules that come out certainly aren't the same ones you pumped into the other end.
