What are the laws on taking a tune and changing the words? Weird Al style, for instance. I won't be selling the new versions or anything, and I doubt they'd even see daylight beyond me. But I'm just wondering because I don't want to start doing something that is illegal and start ruining someone's hard work.

well, according to the man himself, he says there's nothing unlawful about what he does, but as a personal rule, he gets permission. if they say no, he won't do it. he doesn't even have to ask them though.
If you're just doing it for fun nobody cares. If you did anything with the song, you'd potentially be infringing on the copyright of the songwriter (even with your own lyrics, you're using their melody).

I doubt that it's a particularly useful exercise if you want to learn to write your own songs, but YMMV.
I always thought parody works were legal. I don't think it'll really help you develop as a songwriter though, unless you just want to write parody songs.