How can I find my PPSSPP saves in Ubuntu?
The PPSSPP Emulator saves to the memstick\PSP\SAVEDATA subfolder in the PPSSPP directory.
ie. ...\{PPSSPP installed directory}\memstick\PSP\SAVEDATA
If you have built the emulator from source files, it will be located in /home/[user]/.config/ppsspp/PSP/SAVEDATA.
From here you can just copy either the entire folder, or relevant files/folders and then store the copy somewhere else. If you need to restore your backup just move the copy back.