Yes you do, unless you plan to Crossfire or SLI it with your new one, but if your new one is a different kind than your old one, then you shouldn't do that, so just remove the old one along with the drivers.
I've looked inside my pc and it looks like its onboard, how would I disable my old graphics card and enable it later if I wanted to?
You don't. Install the new cord, and plug the monitor into the card, not the motherboard, then install the video drivers for the card.
You can go into the BIOS when the system boots, usually F2 or Del. There should be an option to switch from primary onboard to primary PCIE for the videocard.
It should work without doing that, just plug it in and try it.