To back up saves:
1) Start up the G6 in the DS/GBA, and let it process the save data. When it gets to the menu, turn it off. This step is important; do not skip it.
2) Connect the G6 to the computer and copy the .0 file for the game you want to back up to your PC. GBA saves are in the \GBA folder, DS saves are in the \NDS folder.
To copy saves back to the G6, do step 1, then copy the save back to the appropriate folder on the G6.
You don't need to use the U-disk manager for this, My Computer/Windows Explorer works just fine.