If you cut the cord on the wired controller there are 4 wires: red and black for power, and green and white for data signals. So apparently if you simply hookup up the power and leave the data wires floating you cannot immediately utilize some of its basic parts, such as the potentiometers. Now that I think about it, I guess I should have tried to ground the signal wires to see if the pots and stuff work, instead of leaving the signal wires floating.
What I did was I found the potentiometer solder points and soldered directly to those, providing power and using the signal output. There are potentiometers on the left and right triggers, and 4 total on both joysticks. All of the potentiometers have three grouped pins in a row, and the middle is the signal. I supplied power from the Arduino to the outer pins and hooked up the middle signal to an Arduino input pin.
Supplying the main power to the controller does let the buttons work, however. These are pressure-sensitive touch pads that go low when pressure is applied, so they are pulled up. There is a space next to these pads available for a wire to be soldered. I had to scratch off the lacquer with a razor blade for the solder to stick.
The left and right "bumper" buttons are simple momentary push buttons. No confusion there, just use a DMM to find where you would want to connect.
Using the XBOX controller for its basic components is pretty straightforward, but soldering may be a little trouble because one of the potentiometer's pins are underneath the trigger assemblies.