Do I need resistors or capacitors on this relay? I'm using my Arduino to control a relay switch, which has an extension cord attached to it (see photo).

The relay is connected to the Arduino via pin 9, which is set to pinMode(9, OUTPUT);

Do I need any resistors or capacitors or diodes attached to this? Almost feels too simple, given I'm running AC power through the relay. But maybe no?

It definitely works, but I want to make sure I'm not going to burn something out (or down).
Yes, that is too simple.

The Arduino's IO pin is not designed to directly drive a relay. It needs to go through a transistor first, and you must include a diode to absorb the back-EMF generated by the collapsing magnetic field.

Google for "Arduino Relay" and you will find hundreds of thousands of examples.

In short, the circuit you need to create is this: