After a few days of work (there was a problem with activating the SIM card), I have a fully functional web-based interface to communicate with my modem!

How it works

My setup:

1. Arduino connected to SIM7080G modem module which allows communication of serial port with modem, through human readable chat format (not AT commands).
2. A website hosted by Pythonanywhere of a simple chat interface
3. The modem communicates with my website through HTTP GET and POST requests. 

Here are some images of the setup:

This is the web interface:

Here is the hardware setup

Here is the serial port of arduino:

Here is the link to the code: https://github.com/bolanxu/Terminal-Style-Cellular-Device. (Look at the Web-Based-Chat-Interface branch.)

If you have any questions or comments please comment below!