My question is basically what exactly is electricity? I've simply been told before that it's a flow of electrons, but this seems too basic and doesn't show that electricity is instant. What I mean is turning a switch has no delay between that and a light coming on. Is it really instantaneous? Or is it just so fast that we don't notice it?
It's just so fast you don't notice it. You won't see the effect of the travel time in something like turning on a light, because your eyes aren't fast enough to register the delay, but if you do even moderately precise experiments involving signal transmission and look at it on an oscilloscope, you will find that the travel time is easily measurable. The speed of signal propagation is close to that of light, or about a foot per nanosecond.
(It's worth noting that this is not the speed of electrons moving through the wires, which is dramatically slower. The signal is a disturbance that propagates more rapidly than the drift velocity of electrons in a conductor.)
NO. Electricity is NOT instantaneous.
Light has finite velocity $3 \times 10^8\ \rm m/s$.
Nothing can travel faster than light. This means everything has finite velocity less than light. Nothing travels instantaneously. Hence electrons have finite velocity too.
The speed of conduction of electricity (usually voltage) in a conducting wire is usually determined by the time it takes for the current to saturate the capacitance of the wire or circuit. For example, this is why local processes on the same computer chip run much faster than the same process offloaded to a separate chip, i.e. the circuits have much lower capacitance.
Note that this is completely distinct from the universal maximal propagation velocity of any causal interaction in the universe, as seen by any observer, which also happens to be the speed of electromagnetic propagation in a vacuum, also called "c".
From the point of view of simple models, one can depict "little particles" called electrons - huge amounts of them - moving slightly through a lattice of atoms because of an external cause that created a force field in the material. The movement of the particles is a drift velocity really quite slow. E.g. in a copper wire of 1 mm diameter and a current of 3 amperes, the drift velocity of electrons is approx. 1.0 m/hour (for the calculation see the numerical example).
The velocity of the electric signal through the wire is with the speed of light however. To get the idea, one can depict a tube filled with marbles. When you insert a marble at one end, "instantly" another marble falls out at the other end, however the marbles themselves only shift over a small distance.
This is an explanation from the point of view of a particular (simple) model. Things become more complicated, more difficult or impossible to depict in the context of other models, e.g. pure quantum mechanics. In a pure quantum mechanical model individual hard core electrons do not exist and the explanation becomes purely mathematical.
Picture a 100-car long train at the station, and the engineer manoeuvring locomotive to attach it to the front of the train. As soon as the locomotive bumps into the front car the car shifts backwards slightly and bumps the 2nd car, which in turn bumps the 3rd car, and in less than a second you hear the bump from the 100th car. The signal from the 1st car propagated to the 100th car very fast, and it's true that the cause of the signal was the movement of the cars, but it's not true that the 1st car arrived at the position of the 100th car.
Electricity is sort of like that: the electrons near the switch move very slightly, that movement gets acknowledged by the neighbouring electrons who also move very slightly, etc., and you have near light speed propagation of the signal from the switch to the lightbulb. Although it's true that said signal propagation is due to the movement of electrons, the electrons didn't move at such speed to from the switch to the lightbulb, they just nudge each other a little bit. Just as with the train, the nudging propagated a lot faster then the nudgers and the nudgees.
I don't feel that any of the answers is truly satisfactory. Most merely offer analogies. If I turn on a lamp, the effect does propagate from one end of the connecting conductor to the other almost instantaneously. The effect, however, has not adequately been described as yet by the propagation of electromagnetic fields. Kirchhoff, Sommerfeld, and many others have tried, but no one has explained why twisting, turning, or otherwise changing the path of the conductor has little or no effect on the phenomenon.
One of the early twentieth century investigators (Fleming?) used the analogy of soldiers passing by a reviewing stand. Each soldier moves only a step at a time, yet the overall "throughput" is from one end of the file to the other. But the true physical question is what impels each soldier? Somehow, in the electrical case, the influence seems to propagate along the wire-and (at low frequencies, anyway) not directly through space as per the analysis of Kirchhoff, Sommerfeld, et al.
In fine, the question is still an open one.
Nope, electrons "bump" other electrons in a sort of chain reaction until there is an equilibrium. Try to imagine a limp rope, and then imagine tugging on that rope, and you can see the tug sort of moving down the rope. This is sort of how electricity moves, and it takes time making it not instantaneous, but very fast.