No matter if you are checking a transformer for a doorbell or your central air conditioner, The concept is the same. You will know for sure now if your transformer is working or not.
First you need to get out your test meter, and set it for 120 (240 in some cases, you need to know what supply voltage you have) volts AC. The transformer can be divided into two parts. the supply voltage or high voltage, and the secondary or low voltage. The first thing we need to determine is, are we getting correct supply voltage? The supply voltage wires coming from the transformer can be many colors, the most common for a 120 volt ac transformer supply voltage is black and white. The black being the hot wire and the white being the neutral common wire. First use the black lead of your voltmeter and place the lead inside the wire nut of the black wire of the transformer, place the red lead of the test meter inside the wire nut of the white wire from the transformer. You should read 120 volts ac on your meter. if not then place the red leader to the ground wire, if voltage is now present then you have a bad common wire. If no voltage is present in ether case you have a bad breaker or voltage has become interrupted somewhere along the way and needs to be repaired. If voltage is present then you have supply voltage and now go to step two.
Now we need to check for the presence of our secondary low voltage. the secondary voltage wires are on the other end of the transformer as the primary you just checked. The colors here can also vary, Most of the time they are red and blue, but may be another color. set your meter to 24 volt AC, and place the red lead of your meter in the wire nut of the red wire, and the black lead into the wire nut of the other color wire, If 24 volts is present your transformer is okay. If 24 volts (or what ever rated voltage your transformer is) is not present replace the transformer. be sure to turn off the power at the breaker before you replace the transformer.
If wire nuts are not used on your transformer, then you probably have the wires connected by spade connecters. In which case you will check for voltage right at the transformer.