Clearly I'm not using the analog pins correctly...

Great, thanks. Understood.

Now, the wiper is the one that, in a schematic, is in the middle, right? I am connecting the one on the right to +5, the one on the left to ground, and the one in the center to A0.

Edit: Here is a photo of the pot.

I have red on +5, white on GND, and yellow on A0.