The buttons work valves in a vacuum system that moves doors to rout the air to different vents.
Temperature slider moves a blend door, via cable, that directs 0-100% of the air through the heater core, which is always heated by engine coolant.
The vacuum system is fed from a small tank, white plastic, or older ones might be a black ball shape, over on passenger side under the hood. One small line goes in to the button assembly. You should hear hisses and things move if you change button settings immediately after shutting off the engine.
I don't recall where the contact for the blower motor is, but also be aware that the lower speeds often stop working due to the resistors burning out. They are on a thing down by the motor, in the air path, with 4 wires connector.