From your diagram, you appear to be using an ESP8266-01?
Another ESP8266 AT command library that may be of interest/use.
Note that the AT command libraries are very dependent on the version of the firmware that you have on the device. Depending on when/where you bought the device, this firmware will vary significantly. I've not had any of the libraries work correctly with current versions of the firmware.
AT+GMR at command (using your serial terminal) to determine the firmware version. If you're on Windows/PC updating the firmware to a previous version (the 0.9.x.x range seems to be the most stable according to my research) is relatively straightforward. If you're on Mac, it's not as straightforward...
Official/current release of the firmware is available from Espressif's Github repo.
There are plenty of directions on the net for getting firmware onto the ESP8266 for both Mac and Windows/PC. YMMV as to their accuracy/ease of use.
Note that different versions of the firmware default to different baud rates. Some default to 115200, others to 9600. If you decide to use SoftwareSerial (rather than Serial, which your diagram suggests is what you're currently using) you'll need it to run on 9600. I've not personally tested what Serial supports up to (but my understanding is that you can run up to 115200). Note that there are different commands for setting the ESP8266 AT command firmware baud rate depending on the firmware version/supplier. The official SDK uses the
AT+IPR=9600 command. This stores the baud rate to the device (i.e. you don't need to send it every time, just once and it's then persistent).
If you haven't already, I'd recommend you get the system working with a simple serial terminal program (CoolTerm, for example) before getting into Arduino-land.
Note that the ESP8266 does not reliably run from the 3v3 power rail on the Arduino. The ESP8266 requires up to 300mA, the Arduino is capability of around 50mA to my understanding. Symptoms of insufficient power include random resetting of the device. Best to run a separate 3v3 source to the ESP, or if you have a strong enough 5v supply, use a step-down converter. Your diagram indicates you're using a standard Arduino, which runs 5v out of the serial pins (or digital outs), but your comments indicate you're running 3v3 (which means you're probably running a 3v3 Pro Mini or similar)? If you're not running a 3v3 device, you'll definitely need to run a 5v-3v3 logic level converter (I can't link to examples due to not having enough reputation points—but these are relatively easy to get from SparkFun or similar. I picked up a combined step-down converter with logic level converter which does both tasks.
Lastly, If you don't need the ADC from the Arduino device for your application (which it sounds like you don't given your project description) I'd strongly recommend you reconsider the advice from gmag11 and yeti re: running Arduino on the ESP8266. It is much more stable, and the extra SRAM will make a big difference with processing/building HTTP request strings etc. I wasted a lot of time (unsuccessfully) trying to get AT commands working before switching to Arduino on the ESP8266, which has been a breeze by comparison.