What gives the speed of signals? I've read somewhere that electrons move at 1mm/s in wire, so how is it possible that the signal goes so fast even if the current moves so slow?
The speed through a wire is usually a sizable fraction of the speed of light. Have you seen those executive toys with the balls where you drop the one of the left and the one on the right moves without the ones in the middle seeming to move? It's called "Newton's Cradle" and is a decent analogy to electric signal propagation through a wire.
The electron drift through the wire is slow
The electromagnetic field AROUND the wire travels/propagates at near the speed of light
Its not exactly the speed of light because there is a propagation velocity factor effect of the conductor and any insulation that may be around it