OK a few quick and dirty checks.
With power ON, measure the voltage between base-emitter,
for a germanium transistor this will be 0.2 to 0.3 volts,
for a silicon transistor this will be 0.6 to 0.7 volts.
A darlington transistor will show about 1.2volts.
With power OFF, measure the resistance from base to emitter and base to collector. This test is usually best done with the transistor removed from circuit.
In this test the transistor should look like two diodes with a common connection at the base.
When using a digital multimeter set it to the "diode test" function (using the ohms range probably will not work):
A PNP transistor should show a low resistance from base to emitter and base to collector when the -ve probe of the meter is on the base and high resistance when the +ve probe is on the base.
A NPN transistor should show a low resistance from base to emitter and base to collector when the +ve probe of the meter is on the base and high resistance when the -ve probe is on the base.
When using an analogue multimeter set it to the 10/20kΩ range:
A PNP transistor should show a low resistance from base to emitter and base to collector when the +ve probe of the meter is on the base and high resistance when the -ve probe is on the base.
A NPN transistor should show a low resistance from base to emitter and base to collector when the -ve probe of the meter is on the base and high resistance when the +ve probe is on the base.
Caveats trips and traps.
These tests will not necessarliy find all problems, but will identify a transistor which is busted.
Dont forget to check from emitter to collector, the recesses of my memory tell me that I once found a transistor which checked out OK B-E and B-C but was short circuit from C-E.
Digital multimeters and simple analogue multimeters are different, on the ohms and diode test ranges, the polarity of the probes is opposite.
A transistor is not two diodes back to back, it just looks like that when tested with a multimeter.
JimB