I have worked with GSM modem though didn’t used an external MCU with it but have fair Idea of how it should work:-

1. Get AT Command Manual for your GSM modem

2. Using a command line tool like minicom for Linux and CuteCom for Windows send AT Commands to your modem
[ Recommended functions to test:-

Network registration, data call, MQTT, HTTP POST/GET, FOTA/DFOTA , Low Power Mode, CFUN commands

3. Above steps will make sure your modem works and how it’s failure behaviour and AT responses. Time to do the external MCU part

4. Get UART up for STM and Find out what baud rates would work to send AT Command to Modem

5. Now plan the Architecture for you code, it’s the most important part for embedded system just a good code flow discussion with your team should be good to start with.

6. At this stage you need to code up the worker functions for sending and reading the AT responses to your modem, data handling threads, network threads and any worker functions like for MQTT HTTP etc

Tip:-
If this is a serious project
Then always Expect that your modem is bound to fail and become unresponsive at times.

The two weapons to save your ass:-

1. CFUN command
2. Saving the current states in UFS of the modem and then a planned reboot.