How to indicate a relay is activated?
Both the LED (with a series resistor!) and a buzzer are viable options.

For the LED put a resistor in series with the LED. For 12V and 20 mA the value would eb 600 Ohm, I suggest you start with a 1k resistor. Do add a suppressor diode across the relais coil, any 1N400x  will do.

For a 12V buzzer no series resistor is needed, but I would still add the suppressor diode. The weird effect you observed might be due to wiring the buzzer between the output and ground (instead of between output an +12V).