Can't detect new graphics card

Disable onboard graphics in Device Manager (don't uninstall!)
Shut down.
Plug graphics card in & out.
Boot system.
If detected, try installing the driver.
If not, enter safe mode.
Run a driver sweeper program to clear all traces of AMD driver files.
Reboot and try installing drivers.

Hope this helps.  

As it turns out I found a (at least temporary) solution before reading your suggestion. [ P.S. What AMD suggests is remarkably similar]

For some reason, reseting the bios to their defaults before starting the uninstalling phase allows for a complete install, but then the next boot after installation will fail.

What makes it stranger is that none of the bios change when I do the uninstall OR the install, the reinstall probably does omething in the background, IDK

I've been thinking about getting a boot log and then getting rid of whatever is causing the problem, but I'll probably make do by putting the PC in sleep mode instead of shutting it down until someone suggests a more permanent solution.

Or until I have some spare time and won't mind not being able to play some really good games I just bought

Hmm, sounds like that BIOS might have something buggy. Have you checked if HP has released another update for your BIOS? Flashing your BIOS to a newer version may help as I've found with most of my builds which use newer parts/

Sadly enough, I think I may already have that update since my PC doesn't have any problems starting from sleep.

The curious thing is, even though it has booted successfully with the new drivers FULLY installed, something happens when it starts back up that causes my problem.

The best solution I can hope for is something that not require the computer to boot before the fix is complete

If nothing else, I could reinstall the OS, but I'd like to use that as an absolute last resort.

I really doubt it has anything to do with the OS. But, the best way to really find out is to boot in Safe Mode and see if it works.
By the way, what version of AMD's drivers are you using?