First, a digital multimeter is easier to use.
I would start at the battery where the bilge pump wires start and check the connections there. Make sure they are not corroded, then work your way to where the pump is mounted. If the wires are old a replacement is a good idea - one piece without connections from power to pump or float switch if you have one.
The pump doesn't just need 12V on the live terminal, it needs a ground on the other terminal to complete the circuit.
The best way to test it is as follows :
Connect the red probe to the + terminal on the pump, and the other to a ground. Where to find a ground? The engine, a negative battery terminal, or often the through-hullss are grounded. In volts mode you should get around 12V.
Then set the meter to resistance mode. First connect the two probes together, that should give 0-1 ohms on the display. Good, the meter is working. Now test between the - terminal on the pump, and ground. You should get around 0 ohms. Anything over one is bad.
I suspect you will see infinite ohms because the float switch isn't closed to complete the circuit. So close the float switch and see if that starts the pump and brings the resistance reading down to about zero. I suspect it will not, as the float switch is faulty.
The float switches seem to fail about as often as the pumps, btw.