I don't get how an electric current can go through a conductor faster than the individual electrons flowing through. How can the electrons make a current go across a conductor at the speed of light when the electrons themselves aren't moving across the conductor at that speed?

Say you have a conductor that is 3 electrons thick....

 

A--------B----------C

 

A moves to the right a bit because of teh current, B "sees" this move by interactions using virtual photos, travelling at c, so gets repelled a bit, it starts to move c*distance after A starts to move, so this might be much faster than A is actually moving, it then does the same thing to C, which then also moves....

It may help to draw an analogy from he electric field with the magnetic field. Imagine a tube with a load of cylindrical magnets inside, each aligned the same way, so each magnet pushes the ones next to it. if you push one magnet in one end of the tube, even if you push it slowly, a magnet will come out of the other side almost instantly.

 

There was a class demonstration with a queue of students in a line, and you pushed the one at the back, and the one at the front moved almost immediately as every person in the queue was pushed against the person in front. The deciding factor is how quickly each part responds to the part behind it, and the electrons respond very quickly, even if they don't individually move very quickly.

Have you ever seen the little desktop device with a number of metal balls touching each other, and each suspended by a string? Pull one away at the end, and when it hits its neighbor............................imp

Or, think of the electrons in the wire as being right next to one another so the form an "electron rod". Move an electron at one end slightly and the electron at the other end moves (almost) instantly!