When All Else Fails, Go to the DeLi!
After doing a bit more research, I came across DeLi Linux, a Linux distribution that claimed it could run on extremely old computers, even ones that had Windows 3.1! I had to check it out.
I burned the .ISO image onto a CD, and booted the computer from the CD drive. Surely enough, it worked. This time, no kernel panic errors. The setup was pretty smooth, asking what keyboard language I wanted to use, if I wanted to install additional packages, and some questions about the boot manager (LILO on DeLi Linux). Extremely simple!
So I restarted after the installation was complete, and up came the UNIX shell. Success! Now to test out X Windows. Before using X, you will need to setup an X server, which is also relatively easy. DeLi is very helpful when doing this.