In which direction do electric signals flow?
If electric signals are the propagation of EM waves at close to c in which direction do these waves travel? In the conventional direction? In the direction of electron flow? In both directions?
"Both" directions, as well as an omnidirectional component.
The best example might be an Ethernet cable. A bit is represented as a pulse. To transmit this pulse, the driving chip puts one output low and another high briefly. If it left them like that, then a conventional circuit would be formed with current flowing in a loop between them through the termination resistor on the other end. However, it does not, it quickly (let's say one nanosecond) moves the drivers back to equal voltages.
This launches a rising edge on the positive transmit wire and a falling edge in the negative transmit wire. If you have a fast enough oscilloscope clipped to the wire, these will appear as voltages, but really they're a small EM pulse travelling down the wire.
Each pulse also radiates an EM wave outside the conductor. Because the wires are twisted together at a constant distance, the "positive" wave emitted almost completely cancels the "negative" wave emitted. If this wasn't done, you wouldn't be able to use your radio in the same room due to the EM interference.
The cable can be modelled as a chain of tiny inductors along the wire, with a ladder of tiny capacitors between the two conductors. Each inductor stores the energy of the incoming current change in a magnetic field, before re-emitting it forwards along the wire.
The general subject is called a "transmission line". The effective velocity is below the speed of light, but not much (see "velocity factor").
travel of holes that gets close to the speed of light
It's a bit more complicated than that. (I've not bothered to fully understand that myself!) However, I believe this is why NPN transistors (hole mobility in the P region) are preferred to PNP transistors (electron mobility in the N region).
One of these days I'm going to write a canonical article on why electrons are a distraction for learning about electricity. Electricity is fairly straightforward and follows some mathematical laws that can be analysed with basic calculus. Electrons are not little pingpong balls in a tube, they are weird quantum objects that are capable of appearing, disappearing, tunneling through solid objects, and are much harder to model properly.
signals travel in all directions like waves in a pond, they are a disturbance in the current not necessarily a current themselves.
The signal in a cable is carried by a radiowave which propagates in the space around the wires. Parallel wires or twisted pair have this capability. Coaxial cable also has it, but the wave is limited to the insulation layer between the middle wire and the shield.
The wave happens as electric and magnetic fields, it's not inside the metal. The electrons in metals only make possible that the wave propagates along the cable. The fields of the wave induce some current to the metal surface and we can also squeeze the strength of the electric field to single number, the voltage which, of course, depends on place and time if there's waves. At low frequencies we can virtually forget the waves and make all calculations with voltages and currents. A 60Hz electric power line must be more than 100 miles long before skipping the existence of a wave starts to cause substantial error.
One common practice is to say that wires longer than 10% of the wavelength must be considered as transmission lines which carry waves. If they were not designed as proper transmission lines the result would be unpredictable. I have found in my experiments that 1% or less of the shortest wavelength is a safer limit for freeform wirings.
The wave starts from any change of electric or magnetic field. One can for ex. switch the cable ends to a voltage source (line transmitter ICs do it) or turn the polarity. The wave propagates along the cable further from the point were the change was initiated. Electrons move locally as the fields force. At the same time the current can well be to different directions at different places of a wire. That doesn't happen with continuous unchanging DC.
ADD due the inserted battery+ switch+ 1km wire + Led+ Resistor+ 2 km wire example.
The geometry of the wires and how they are placed over the ground affect radically how the signal reaches the LED. If you reversed the battery and the LED and tried again no difference would be exist, the polarity doesn't affect the wave. This isn't a transmission line if the wires are not parallel. It's not possible to calculate nor simulate the case without knowing the exact geometry. If the wires are wounded to coils they are inductors which surely has some effect in the circuit. In a lucky other geometry case you emit a measurable radiowave to space because you can have a working antenna.