The black cable wire is connected to the top terminal on the switch and spliced to the black and blue wires at the fan at the other end. The neutral source and ground are spliced through to the white and ground connections at the fan.
This page contains wiring diagrams for ground fault circuit interrupter (gfci) receptacles. Included are diagrams for multiple gfci’s, a protected standard duplex receptacle, and a protected light fixture. Wiring for a switch and gfci receptacle in the same box is also shown to protect the switch and light or to protect other receptacles in the circuit. To wire a gfci circuit breaker see this link and wire a gfci switch combo at this link.
Here the exhaust fan is controlled by a timer instead of a switch. There should be two hot wires and a ground coming out of the timer casing, splice one of these to the hot source. Spice the second to the black, cable wire running to the fan. Splice the source neutral to the white cable wire and the ground to the ground wires. At the fan splice the wires, matching the colors of each.
To wire a 3-way switch circuit that controls both the fan and the light, use this diagram. As with all 3-way circuits, the common on one switch is connected to the hot source wire from the circuit. The common on the second 3-way switch is connected to the hot wires on the fan/light.