...and the GRN should be connected to DTR but I have no idea how to handle this signal.
I've not used one, but from the looks of it the GRN pin is attached to the reset pin of the ATmega. By pulling this pin low, it will reset the OpenLog.
You can probably leave it unconnected (it has a pullup to 3.3V), or connect it to an output IO pin.

You would connect the module to the UART serial port of the pic and send the same commands you use in the terminal.

BTW, the Openlog is a 3.3V module, so you would have issues directly connecting it to a 5V uC or FTDI cable.