So I disconnected everything and just plugged in the PSU (on my current computer, not another one). It seems to power up everything fine even after numerous attempts but there are no lights in front of the case and after I plugged in my video card and monitor, there was nothing on the screen. I also noticed that some lights on the motherboard that would normally be lit when the computer is functioning properly is absent.
This is isn't exactly related to the problem but I tried cleaning out the dust on the spinner of my CPU fan: Newegg.com - ZALMAN VF700LED - ALCU 2 Ball Blue LED VGA Cooling Fan with Heatsink - Retail
and so I noticed that the spinner came off kind of easily so I assumed that it was built so it could be taken off and cleaned. Inside of the spinner seems to be a magnet which connects to the small metal ring at the base of the fan. Anyway, I tried to put it back on but no matter how hard I pushed the spinner onto the base of the fan or try different positions, it will not secure itself properly. I can't really tell but was there maybe thermal paste underneath the spinner? There's a lot of gray inside there but I can't exactly tell if that's just the magnet or thermal paste.