Several months later…
A brief update:
I chose the "slow boat from China" shipping method. So the modules didn't arrive until late summer, and then I was busy with other things until a few days ago. I bought both the ESP-01's and the WEMOS D1's as recommended, but I ended up doing the development on the simpler ESP-01. The principal difference is that the ESP-01 is serial only, and the WEMOS is USB. Since I have a few FTDI adapters on hand, there was no trouble interfacing between my computer and the ESP-01. Initially, I worked with the pre-flashed AT command set that the ESP-01 came with. This was very handy for initially getting things communicating and proving out both the serial and wireless end of things. After that, I found an Arduino example project that was close to what I needed. So I modified that, and after a minor amount of debugging, it's now doing exactly what I wanted.
This was my very first Arduino project, and I don't have much C programming experience. So, I was a bit apprehensive. However, things went surprisingly well. If I had any complaint, it would be the lack of a good single source of documentation online. There's a tremendous amount of incomplete and incorrect ESP8266 documentation out there, and I had to sort through a lot of it before I got to the point of connecting the hardware and starting the programming. Even so, it all turned out well, and I'm already thinking up some new projects. So, thanks everyone for the suggestions to use the ESP8266.