Thanks for all of the help everyone has given me. I will try to answer all of your questions
Some specs:
RAMPS 1.4 w/ LCD
12" Heatbed w/ built-in relay
12v 30a power supply (RIP)
14awg wire for power connections (RIP, maybe)
Tests I've done:
Disconnected everything from 30a ps and plugged into power (ps still wouldnt turn on)
Connected old 20a ps to stock RAMPS (ps wouldnt turn on, but worked by itself afterwards)
Connected old RAMPS to old ps using my stock 14awg power wiring (neither would turn on)
Conclusion:
I think that my power wires are bad or broken. I am replacing them with some thicker 12awg wire and testing that now.