UART weird characters

Common issues that came immediately to mind:

1. Ground connection: you need a shared ground to get a usable communication, this gives very weird issues like the ones you describe
2. Unstable clock on either side: as UART has no clock sharing mechanism the clock must be fairly well defined and may not drift much. If you are running from an internal RC oscillator this might be a problem. Better use a resonator or oscillator.
3. Wrong signal levels: Make sure both sides are working with the same signal levels and you are not trying to feed a TTL signal to a RS232 port.
4. Polarity setting: define inactive high/low on both sides the same (normally you won't get anything useful if this is wrong)
5. Correct communication parameters otherwise: baudrate, number of bits, parity, number of stopbits, flowcontrol
6. RX / TX mixed up: Often the signals are called RX and TX for receive and transmit. It also often happens to just connect RX to RX, but it should be RX to TX (receive on one end is transmit on the other) (can also happen the other way round, depending on how people look at it, it's a mess)

I am not sure fully about how to implement this "forwarding" (copy like)

An (often quicker) alternative to this type of software "forwarding" approach, is to monitor the two signals you mention, directly via hardware.

I'm short of time, so excuse the lack of a diagram at the moment. In brief:

MCU Tx -> GSM module Rx -> Here connect Rx of UART-to-USB adapter 1
MCU Rx <- GSM module Tx -> Here connect Rx of UART-to-USB adapter 2

Note: You only connect the Rx inputs of those two UART-to-USB adapters. Leave the Tx pins of those adapters unconnected.

Then you can monitor (and, although I've forgotten the name at the moment, I'm sure I've used software to merge or view) these two data streams on a PC. You could even run two simultaneous copies of "Tera Term" as you suggested, one opening the PC COM port for UART-to-USB adapter 1; the other opening the PC COM port for UART-to-USB adapter 2.

This avoids adding complexity at the MCU now. You can add this MCU functionality later, if you need to and when you have learned those programming skills. However during development, using hardware monitoring can be a quicker way to achieve the end result (i.e. monitoring those serial data streams) which you want.