MQTT + ESP8266-01 and AVR
If you just want MQTT on ESP8266, probably the easiest is to use NodeMCU, a prebuilt firmware which runs lua scripts and includes mqtt support. You'd have to learn a bit of lua, but it's not harder then c++ :)


How to flash NodeMCU to an ESP: https://nodemcu.readthedocs.io/en/dev/en/flash/
Sample of using MQTT on NodeMCU: https://www.cloudmqtt.com/docs-nodemcu.html


Alternatively use the Arduino IDE + ESP8266 core, which has MQTT libraries available for it.

Or last option, use esp-open-dev with tuanpmt mqtt library: https://github.com/tuanpmt/esp_mqtt This is the hardest.

NOTE: All these options have you writing your own code on the ESP8266, but it can be a small and relatively easy amount of code :)

This question is probably better put on Stack Overflow or Arduino Stack Exchange.