The starting point for this is getting yourself an Arduino. The Arduino (known as a "Genuino" outside the US and UK) is a single board computer. It doesn't run Windows or MacOS or Linux.

The Arduino IDE is used to program an ESP8266, and there are plenty of MQTT sketches out there that may be used for this purpose.

As penance, I’ve decided to document the process and share it with all of you, and then also make it work on an ESP8266 board running the Arduino development environment.