How much RAM would I need to upgrade on a new M3 Macbook Pro?

Check the recommended processor and memory configurations for the apps involved.

For the two games cited:

https://us.battle.net/support/en/article/76459

https://na.finalfantasyxiv.com/system_requirements/

For storage, check the recommendations again, and figure you'll need more than that. And storage usage only goes up, absent scattering stuff around, or absent the time and the discipline needed to delete. And figure everything only gets larger over time; your own usage, macOS and apps, everything.

It appears FFXIV requires Rosetta too, which means it's at least partially Intel and not native Apple silicon, and thus translated. Which usually means things can be slower, pending translation.

Since you want to run all of that together, the requirements will add. (And remember that macOS and all of the other apps active also need resources, or need to become inactive and be swapped out.)

Check with a local Apple Store (assuming you have one), and see if they'll let you run the games on an M3 in the store.

But probably your best bet is to ask for guidance in technical forums for the games; at the game vendor forums, or at Reddit, or elsewhere. Guidance from folks that have run these games.

You'll also need a hard disk drive of two to three times the total internal storage for Time Machine backup, either directly connected or configured as network-attached storage.

If you purchase from Apple directly, you typically have 14 days for a full refund. Which means load it all up and test it out right quick, and return it if not.