OK, that sounds good. It's the fuel pumps pressurizing the system.
When it's more of a buzz than a whir, that might mean the pump under the car has been replaced with an aftermarket non-Bosch item or the pump in the tank has failed (especially audible when the tank is less than half full). It's not related to your starting problem, but a thing that needs to be checked later.

Right now, your symptoms seem to point to a failed iginition switch. This should be easily confirmed by measuring the voltage on the service socket hanging there on the firewall (more on that here).

In the mean time several other failure modes have been suggested in this thread ranging from broken igintion switches, failed solenoids to corroded connections.

The service socket sits right there in the middle. Measuring voltage on it with respect to battery negative terminal while twisting the key to III you might see:
- battery voltage: the solenoid should engage and the starter motor should turn; if not, expect an intermittently failing solenoid or corrosion on contacts.
- no or very low voltage: would point towards a failed ignition switch.

With the gearbox in neutral, if you run a wire to the service socket and touch the positive battery terminal momentarily, it should make the starter motor run. Basically, it does the same as shorting the two terminals on the solenoid together albeit in a more convenient and safer way. Repeat this several times to see if the starter motor runs consequently. If so, that would point towards a problem upstream of the service socket (likely the ignition switch). If not, downstream, i.e. contact corrosion or the solenoid/starter motor.
Good luck!