Awesome! Thanks for the replys, everyone.
As per a few of your suggestions, I took it off to play with it.
I seem to remember that some Suntours could be mounted so that they had no tension. The fix was to remove it and wind it up then put it back or something like that. It's been years since I've seen it.
Yes! Thank you, you are right. Once I took it off, I loosened the whole thing, and swung the arm up around (360 degrees) and bam! It wants to go back to how it was before--aka it's providing tension now. I put the whole thing back on, and there's proper tension now!
But when it was off, I was messing with the screws. Perhaps because of that, or perhaps because it's bent or another problem, my shifting's all messed up. For the most part it just needs some adjusting, I think, but when I adjusted it to the point where it would actually get the chain onto the highest gear, it started to hit the spokes when I went into that gear. So I think it might be bent?
I'm going to try to find another Suntour on eBay, but for now, would it be safe to ride it without shifting the back gears, just for a few days? Just adjust it so that I can shift it onto the mid gear, and then not go up or down? Or is this a bad idea?