So the other day I read about the "hold the front brake and rock the bike" test for a loose headset, and as it turns out my headset seems to be a little loose--there's a definite clunk when I do the test.

I've been trying to do all the maintenance myself, so as to learn more about the bike, so I'm going to get a couple of the proper wrenches (it's a threaded headset) and tighten it myself. However, and this is the actual question and reason for the post: is this a big problem that should be addressed right away, or can it wait? I don't notice the loose headset when riding, but I just want to make sure I'm not going to make the problem worse or cause an accident by ignoring it.

Yes, it's a problem that should be addressed right away. If you don't have the tools, it's an inexpensive adjustment at the LBS.

+1 on taking care of it now. Also, I'd suggest opening it up to check/clean/grease the bearings as you been riding it loose.