It will depend on which game you’re playing.
If you’re using Retroarch, from memory it saves save data to the respective game’s directory.
If you’re using a standalone emulator, ie not Retroarch, if sometimes stores the save files in the ~/apps/emulators/ directory.
N64 emulation stores it in a hidden directory in ~/.shared or something go that effect. (Although I’m guessing you don’t have that running yet)
PCSX has changed its directory to ~/apps/emulators/.pcsx/ (another hidden folder) where it holds data for memory cards. If used to be in ~/launcher/sys.py/pcsx/ or somewhere to that effect.
The save file for Cavestory is in ~/games/Cavestory/ or somewhere to that effect.
Your games are all in ~/games/ within subdirectories.
Bios files are in ~/apps/emulators/
Any other games that are built up will need to be double checked manually. Eg, quake has all of its files in a hidden directory within ~/ , so you’ll have to work it out on a per game basis.
In general, backup the following:
If you want, post exactly which game you need backed up, and which emulator you’re using.