The two terminals you need to do all your checking on are the two that in the socket that line up with the terminals that are in the vertical position in your pic. #3 and #5
One is 12v supply that comes from the fuse.(#3 on the relay) With a good fuse, there should be 12v there.
{on a side note, just because there is 12v there does not mean the (entire) circuit is viable enough to carry a load (amperage) Corrosion inside of a wire or in a connection can become bad enough to not allow the circuit to carry a load, but good enough that is will carry 12v for a meter to measure.}
The other connects to the fan. (#5 on the relay)
The diagram shows how the relay electrically sits when it is off. The "bar" flips from 3-4 to 3-5 when it turns on, feeding voltage to the fan.
check between the socket for pin 5, to ground with the fan disconnected. should be infinite. If it is not, there is a short to ground. That is no bueno. that needs corrected.
You can jumper the two socket terminals with a piece of 14ga wire... with the fan disconnected, nothing should happen. If the fuse blows there is a short. That's no bueno, that needs corrected.
Now, if all is well so far,
Witht he jumper wire installed, connect the fan connector back together. it should turn on.
If not, either the ground is bad or the fan is bad.
If the fuse blows, the fan is bad.
To check the ground disconnect the fan. on the body side of the connector check the black wire with the ohm meter to ground. should show .5 ohms or less
If the ground checks ok, the fan is likely bad.
If you get infinite, that is no bueno and needs to be fixed.
You can check the capability of the circuit to carry a load, you can use a headlamp and connect it with wires to the body side of the fan connector. With the jumper wire installed it should light. if it is dim or doesn't light, then the circuit has corrosion somewhere. that is no bueno. It needs to be fixed