ieee488:
if you replace the ESP8266 module's default AT firmware with ESP8266 Arduino Core firmware
ESP8266 and the Arduino IDE – Martyn Currey
Of course, re-programming the ESP8266 flash is a challenge as well
ChrisLamont, it may not be clear to you from the above post that you may not actually need a 'basic' Arduino at all; just an ESP8266 module (like this one) could easily do all you need unless you need a lot of i/o. With that device, you won't need to worry about sending commands from an arduino board to the ESP8266 (or reprogramming the flash; the IDE will do it all for you); the ESP8266 is a microprocessor in it's own right (also has a lot more memory than most Arduinos). And there are plenty of examples of sending data from ESP8266 via wifi to a web server.