ok, start with basics.
Antenna connected.
No code required.
No SIM card.
Power on 5V to Arduino & shield
Check modem red light is on. (5V present)
Hold power button on modem for a second
Check bluebOED comes on (modem power)
Power off.
Insert SIM card...
Repeat above.
After several (up to 20) seconds, blue LED should blink — modem has found a network to connect to...
(doesn’t mean you have SIM credit)
If you get here, check your serial connections to the modem.
(Use a different serial port than the Arduino USB D0/D1 )
There are links on the modem to select which pins are used.
The serial speed I found in all my SIM5320 has been 4800 by default.
Let’s see how you go from here.
If still not working, and blue LED is blinking, write a simple pass-through program from the serial monitor to the modem serial, and vice-versa.
Then you can try simple AT commands without FONA in the middle.
Once that works you can start again with FONA.
P.s. I use the Arduino / modem combination with an external 5V supply.
The Arduino regulator probably can’t cut it to supply the 5V for the modem. (which btw from experience draws most current when finding a network.