I decided to take the plunge and invest a pair of inexpensive studio lights. The whole kit cost less than my portable Canon 580 EX II hot shoe flash (I italicized portable because that flash by itself is twice the size of most point-and-shoot cameras).
The studio lights arrived today. They run on AC power and put out more light than a typical flash, so I'll be able to work with faster shutter speeds and smaller apertures indoors.
I'm no expert at lighting, so it's going to take some practice.
Or maybe not.