I found an old video card around my house last night, so I swapped my current one out for it. I was able to boot my system, but upon entering Windows, I still see graphical trash. That indicates to me that the motherboard is most likely to blame.
After doing a little bit of hardware research last night, it appears that my CPU is still among the best, so I doubt I'll replace that after all. And seeing as my graphics card might not actually be to blame, I'll probably hang on to it as well (it, too, is still fairly decent). The motherboard definitely needs to be replaced, and I'm thinking about going to DDR3 memory instead of DDR2 (though if I stayed with DDR2 I could get by with just purchasing a new motherboard).
So, long story short, the situation doesn't appear to be as dire as I had initially thought. It still bites that I have to deal with this though. Why can't technology just work?