So, I ordered a bunch of parts and put them together. They should all be compatible.
My power supply unit is not faulty. It works fine when I plug it into my other system. It does not work in my new system. The PSU from my old system does not work in my new system either.
Everything is connected correctly. POWER SW, RESET SW, 24 Pin, 8 Pin. I've checked, double checked, triple checked, and repeated that process roughly 20 times.
The only other thing I can think of is that it may be due to my motherboard not being fully secured to the case. The case gave me a diagram for ATX boards and MATX boards (My board is a micro atx), so I put standoffs in all the MATX holes. However, two of them to the far right did not have a matching screw hole on the motherboard, so I removed them from the case and left the holes empty.
Other than that, I'm completely clueless.
ASRock MB-Z68MU3 LGA1155 MATX Motherboard
G.SKILL Ripjaws Series 8GB (2 x 4GB) 240-Pin DDR3 1333MHz DIMM