The best way to find an ancient arfifact is with the Spanish because they start with the knowledge to build Galleons, an improved version of the Galley. Galleons can move in deep waters and can move more squares per turn then the Galley.
Even if you play on Deity the enemies will never, ever get the technology for Galleys or even Galleons before you build your Galleon with the Spanish.
You can also get a random Galley gift from villages if you enter them or by capturing a Barbarian camp.
Just let your city grow until around 10 and you can quickly build a Galleon, if your city has lower population they Galleon will take looong to build. There are mostly around 3/4 artifacts per match so it shouldn't bo too hard finding one this way.
99% chance artfifacts are on little islands in the water.
The artifact of Atlantis is always in the match, in deep water so you will always get it with the Galleon doing it like this.
At the end of this video (3:50) it shows an Artifact on a little island.