What does cerulean mean?
The Cerulean Cave is in Cerulean City. You have to surf from a place around Cerulean Cape down to the Cerulean Cave. The Cerulean Cape is north of Cerulean City across the bridge. Go to the north end of the nugget bridge then left down the grass and follow the mountain to it. Btw you'll need flash and you can find Mewtwo in it.