How are electric and magnetic fields able to travel through space and wires, and how are they more than mathematical constructs? Once I was taught that light is physically made up of in-phase E and B fields oscillating perpendicularly, it was a little baffling because I always thought fields were just analysis tools (or even just a helpful perspective) for the direction/strength of forces for a given charge / charge moving. This kind of goes into transmission line theory in electrical engineering I suppose, but when a change in voltage occurs and propagates through a medium at c... what exactly is the entity--which carries a change in voltage--that travels? Should it just be accepted at face-value by students that electric and magnetic fields are physical things, in real life, that move at the speed of and also in fact are light?
Also, considering an electrical signal traveling through a wire medium: is the electric field in the direction of the current/signal? The answer to this might seem obviously to be yes; however, if electrical signals  are carried as light, then by the directional definition of E x B = wave propagation direction shouldn't the electric fields (and magnetic fields) be perpendicular to the wire / direction of the signal? Or are these EM-wave-intrinsic E/B fields different from the electric field that relates to the current's/signal's direction?
Thanks,
-A curious and constructivist EE major, who hopes to someday understand electricity electron-by-electron
 A: Lets make clear from the beginning that Physics theories use mathematics as a tool, imposing extra axioms called postulates, principles,laws so that those mathematical solutions are picked that fit measurements and observations and also, very important, predict new phenomena which become validated.
Electricity and magnetism were slowly defined and measured  and the mathematical theory that finally conjoined electricity to magnetism and predicted electromagnetic waves was Maxwell's equations.
The equations have inherent in their formulation that the velocity of light in vacuum is c and the Lorentz transformations, which became special relativity. So formulating them was a big step in the history of physics theories.4

Should it just be accepted at face-value by students that electric and magnetic fields are physical things,in real life

Yes, they can be measured and experimented with in the lab.

that move at the speed of and also in fact are light?

In certain boundary conditions in solving Maxwell's equations it can be shown that light is formulated mathematically as sinusoidal waves of electric and magnetic fields:

It is an adequate mathematical description that predicts observable effects, as interference phenomena for example.
To continue with the modeling, classical electrodynamics is a successful model for the behavior of light, electricity and magnetism at certain classical dimensions. It fails to fit data at the quantum level dimensions, where quantum electrodynamics is needed.
The Michelson Morley experiment showed that light does not need a medium to travel on. At the quantum level, light is composed out of zillions of photons, their wavefunctions superimposed quantum mechanically. This double slit single photon at a time experiment shows this clearly . 

Also, considering an electrical signal traveling through a wire medium: is the electric field in the direction of the current/signal? 

The microscopic behavior cannot be analyzed in the classical electrodynamics . No it is not light in the wire. It is the electromagnetic interactions of the electrons and ions in the lattice of the wire, and one needs the quantum mechanical formulation for how energy is transferred with off mass shell photons, if one wants details. It cannot be explained with handwaves. So the model you have in your  question is not viable.
Classically a signal through a wire is just variations in the current going through the wire.  Have a look at this question and answer. 
