I'm going to be putting in a new motherboard, processor, memory, and graphics card.

What is the best way to go about upgrading my Linux distribution to accomodate my new hardware?

I could go the lazy man's way and just buy a new Red Hat CD and have it upgrade everything. Is there a better alternative? I'm not even sure what will happen when I drop in a new motherboard and processor. I'm certain Linux will not be happy when I drop in the new graphics card.

Keep in mind I am hardly a pro with Linux, and I will become pissed off after a few hours of fighting to get something to work.

It will be enough to make sure your kernel will go with your new hardware (if you still have the kernel that was with your distro, it will work).
Graphic card: may be problem with X. You'll need to configure it before starting.
Changing hardware in Linux doesn't mean reinstallation. The worst thing that might happen is kernel recompilation (if you have higly customized one) and changing X configuration (Xconfigurator, for example).

mara's pretty much right - if you haven't recompiled your kernel, then you still have the stock one from red hat - and it'll take pretty much whatever hardware you throw at it. that is, assuming the hardware isn't too new. worst comes to worst, you'll need to upgrade the kernel to accomodate some of the newer hardware. if you get a decent graphics card, then you'll probably want to look around for drivers from the vendor. nvidia has some great drivers available, and ati is also well supported under linux.