Is a signal traveling through fiber faster than a signal traveling through copper? Does the transmission medium affect the speed of a signal? For instance does light traveling through a fiber cable get a bit from $A \implies B$ faster than copper can transmit a bit the over the same distance?
 A: The medium does affect the speed of propagation, but I don't have detailed information on the technologies currently in use. 
The rule of thumb is about 2/3 of the speed in vacuum for either coax cable or fiber. No idea for twisted pair.
A: Well, definitely yes, i think. Depending on properties of the cable as well as the Relative permittivity(also measured as dielectric constant) of the material surrounding the copper is a factor.
EDIT: The velocity factor also called wave propagation speed or velocity of propagation (VoP or vP), of a transmission medium is the speed at which a wavefront (of an acoustic signal, for example, or an electromagnetic signal, a radio signal, a light pulse in a fibre channel or a change of the electrical voltage on a copper wire) passes through the medium, relative to the speed of light. For optical signals, refractive index is a similar quantity.
Twisted pair copper cabling are known to transmit signals at speeds anywhere between 40% to 70% of that of light. So clearly, there are a lot of variables that come into play here that vary with material used, methods of cabling etc. See this table for reference.
As for your question, a quick wikipedia lookup for fiber optic cables reveals this..

The typical value for the cladding of an optical fiber is 1.46. The core value is typically 1.48. The larger the index of refraction, the slower light travels in that medium. From this information, a good rule of thumb is that signal using optical fiber for communication will travel at around 200 million meters per second.

Which at first glance, might seem like faster but does not necessarily mean "always faster" than any/all types of copper cables.
A: Summary: A good rule of thumb is roughly 2/3 the speed of light in a vacuum for all these transmission media. Optic fiber is slower than some and faster than others.
Details:
Electrical energy normally doesn't flow inside of metals.
Poynting-flow diagrams show where the energy flows, and the path(s) traveled by the information carried by that energy, although they are a bit difficult to draw.
As conqenator pointed out,

The refractive index ... Typical values for core and cladding of an optical fiber are 1.48 and 1.46, respectively.

which gives a wave propagation speed of 1/1.48 and 1/1.46 of the speed of light in a vacuum (0.67 c and 0.68 c)  in these regions of a typical optical fiber.
Various popular cables have a wave propagation speed that ranges from 42% to 95% of the speed of light. In particular, 100 Ohm Cat5e cable has a typical propagation speed of 0.64 c. The 300 Ohm Twin-lead has a typical propagation speed of 0.82 c.
dmckee reminded us that it's usually adequate to approximate all of these as around 2/3 c.
So if you use Cat5e cable, bits will arrive slightly later than bits sent through fiber optic. If you use 300 Ohm Twin-lead, bits will arrive slightly before bits sent through fiber optic. 
A: Depends on coaxial cable's dielectric RG58 using PVC will have a Velocity of Propagation Factor(VOPF) of .66 Andrew's FSJ1 having foam dielectric has a VOPF of .78 Fiber Optic has a VOPF of .68. A pulse sent down 100' of RG58 will have 154,045pS delay, FSJ1 will have a 130,346pS delay, Fiber will have a 149,515pS delay
