What should I set the multimeter for testing the transformer and should I test the contacts or the wires going out

That depends on what you intend to measure...
Always first measure the input voltage (120 to 240 Volts AC - so use the 250 to 400 Volt AC range).
If there is input voltage, than check the transformer for the intended output voltage (which depends on the circuitry connected to it). One you know, set the multimeter to that voltage range and measure the output voltage.

If you want to test the voltages to and from a transformer, you test on whatever is available. Some transformers have connectors or connector strips. At least it should be easy to measure the input voltage.

:)