So I replaced the screen on my iPhone 4 with no issues using the iFixit tutorial. When I turned my phone back on it worked for about 20 minutes, and then it turned off and started acting funny. It would only work if plugged into a wall outlet, which immediately made me think something had happened to the battery (the phone had no power issues before the screen fix).
I ordered a new battery (overnight shipping), and replaced it the next day in about 15 minutes. Having replaced the screen the previous day, I knew what I was doing, and it was an easy fix. The phone powered right up and I've got a working phone again.
This may be more of a note to the screen fixing part of the tutorial, but I would just make sure to put the back cover on the phone before powering back up. I have a feeling that the grounding clip under the battery may have been the culprit with my perfectly good battery. I have a suspicion that powering back on my phone before replacing the back cover may have short circuited the battery. Watch out!!