If you can use an ohm meter, you should be able to determine whether a transformer is good or bad.
Put the ohm meter on its lowest scale, like 100 ohms scale as opposed to 100K ohms scale. Touch the leads of the ohm meter together and verify the reading is zero ohms, adjust variable knob to make it read zero if it is not already on zero. (If the battery is too low to bring the needle to zero and reads 2 ohms for example, take note and call that zero for this test.)
A transformer consists of 3 or more coils of wire. 2 yellow wires for example represents the leads of one coil. 2 brown wires represents the leads of another coil. You can do a quick test for each winding for an open while the transformer is still connected in a circuit. Assuming you're using a cheap inaccurate ohm meter. Look for a reading of somewhere between one and about 10 ohms. If any winding reads higher than 10 ohms you have probably found a bad transformer. Unless you didn't get a good connection to the coil leads with your test leads. Always check at least 3 times before you draw a conclusion.
If you didn't spot a bad transformer with the quick test, you'll need to unplug or unsolder the leads of the transformer from the circuit. Now measure each coils resistance again. With a better than cheap ohm meter expect all windings to read between one and five ohms.
Assuming all coils passed the open test, test each coil for shorts to other coils. For example put one test lead of your meter to one brown coil lead and touch each of the other coils leads with the other lead of your test meter. Expect a million ohms reading to verify no short was found between any 2 coils.
Repeat the test of step 4 with each coil to verify no coils are shorted to any other coil. Any zero resistance readings suggests a short and bad transformer.
If you're working with a battery powered vehicle learn safety first, they've got more than just a 12 volt battery to watch out for.