A treatment of basic Kaluza-Klein theory I'm looking for a treatment of the original basic Kaluza-Klein theory. 
Can someone recommend a review article or something?
 A: How's this: http://www.weylmann.com/kaluza.pdf ?
Be careful though, because there are a couple very big errors that the author makes. For example in equation (6) there should not be a factor of 1/2 in front of the second term on the right-hand side. It should just be a factor of one. If you do the Christoffel symbol calculations and expand out the 5D geodesic equation, you'll get the following:
$$\frac{d^2 x^\lambda}{ds^2}+\begin{Bmatrix} \lambda \\ \mu \nu \end{Bmatrix} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}=-k\left ( \frac{dx^5}{ds} +A_\nu \frac{dx^\nu}{ds} \right )F^\lambda_{~\mu}\frac{dx^\mu}{ds}$$
In fact, the factor 1/2 that the author has in his paper keeps the equation from being gauge-invariant. Similarly, after equation (6) he says:

This expression is now fully covariant, although the $A_\nu F^\lambda_{~\mu}$ term does not have any classical correspondence.

This is incorrect. It makes no sense to split up the terms, which is why I factored them like you see above. It actually turns out that Noether conserved momentum about the curled-up 5th dimension is:
$$-mk\left ( \frac{dx^5}{ds} +A_\nu \frac{dx^\nu}{ds} \right )$$
So it's a conserved quantity which is gauge-invariant. By comparison with the usual Lorentz Force Law, this conserved momentum is therefore associated with charge.
You should also note that in more modern treatments of KK theory $g_{55}$ isn't restricted to being a constant, and is promoted to a scalar field. Physically this means that the 5th dimension is allowed to have variable size.
