Thanks Grumpy_Mike. I saw that you posted similar advice here but didn't appreciate it.
I still find this very confusing, and Google/Wikipedia aren't a great teachers. 
It sounds like I should be hooking up 120 VAC to the speaker, using an NPN transistor and a strong capacitor to control with the Arduino. I will have to look a lot more into "push-pull circuits" because I don't want to screw up 120 VAC.