Before you installed the 560 (nice choice), was your onboard video working OK?
If so, remove the EVGA and reboot and go into BIOS and make sure you have the appropriate PCI-e slot enabled and it's seated in that slot.
And no, no video was displayed at all, the screen was completely dark. I think I did two things wrong, one is that I don't have the right power supply (thanks for the link, that one is 500+, the one I have installed is Bestec 250W, not nearly enough I see) and the other was that I didn't follow your advice and go into the BIOS and enabled it, so I think I'll do that after I pick up a new power supply.
Noob question here, but is it necessary that I remove the old graphics card? Not going to lie, I have no idea where it is.
Looks like you were using onboard graphics. That means the chip is part of your motherboard and cant be removed. Good news is, you dont have to remove it, Once you put a card in the pcix slot your bios should switch from the integrated graphics to the card