So, basically I feel like an idiot. I found this and thought I'd try it (the toilet only started running recently). Looks pretty simple, right? I thought I could handle it.
I got all the way through it. I replaced the part. I connected the hose back up. Then I turned the water back on. And that's where my troubles started.
The thing that lets water in? That started dripping. A lot. And then the place where the hose connects to the toilet was dripping too. So, I turned off the water, tightened everything, and tried again.
In the end (after several attempts), I got it so that it wasn't dripping much. But it was still dripping. And rather than screw it up any more, I figured it was time to call a professional.
I'm not sure what lesson I'm taking away from this.