Feathermerchant

The diode is circled in green. The end with the band should be toward the relay.
I'm guessing that a 500mF 20V (or more) capacitor should be fine.
The resister is not needed. It is labeled "10K fixed resist" in the picture.

The Machine

Sweet, I've got a couple diodes on one of my switches that turn on a few other switches simultaneously so I can handle that part. My new question is this: is the capacitor wired in like the diode or does it get grounded somehow like in the drawing? Thanks again so much you guys, I'm learning a ton here!

Feathermerchant

As in the schematic, The + lead of the capacitor is connected to the banded end of the diode and the - end is grounded. The capacitor will be marked typically only on one lead.