Ah, never mind. I installed the latest version of Java and now it works. I'm not very happy with the overall experience described above, though. The previous version of Mathematica worked fine on my computer before I installed the new version, so it's surprising that I would have to update Java. Even so, why was there no message telling me that I should update Java? That would have been very helpful, and would have saved a ton of confusion and Googling.
Also, on a related note, I am getting another issue that I've gotten before in a previous version of Mathematica. I believe there is also a thread in the forum where another user has the same problem. For the very first command, even if it's something simple like "1+1", if I try to evaluate it, then it hangs indefinitely. I have to go to Evaluation -> Abort Evaluation to get it to stop. Then everything works fine after that. This seems to be a mild annoyance, but I wonder why it hasn't been fixed. The other thread seemed to suggest that this has something to do with the suggestions bar though.