Would a card with no serial number and a notation on the bottom saying "first edition" be real or fake?

It depends - some cards like the Egyptian God Cards (Slifer The Sky Dragon, Obelisk The Tormentor, The Winged Dragon of Ra, The Winged Dragon of Ra - Sphere Mode, The Winged Dragon of Ra - Immortal Phoenix) do not have serial numbers. This goes for the collectible and playable versions of the Egyptian God Cards. But if it's a card like Blue-Eyes White Dragon, it most likely is fake. Other cards that don't have a serial number include Black Luster Soldier (Ritual monster), Five-Headed Dragon, etc. This does not mean they're fake. Sometimes cards are accidentally printed without serial numbers and are misprints.