Since sam hasn't posted an answer, I'm going to guess that this is the Balance Keepers trilogy (2014-2016) by Lindsay Cummings.
In the first book The Fires of Calderon the hero, Albert Flynn, and two of his friends, Birdie and Leroy, are led on an adventure under the Earth by a dog that bears a letter from Albert's missing father. From the Goodreads entry:
After following a mysterious map into the woods and then under the woods, eleven-year-old Albert Flynn learns he’s a Balance Keeper—someone with special magical skills for fixing problems in three underground Realms at the Core of the earth.
This review describes how they draw Tiles from a fountain, and each Tile is white with a black rune and grants a specific power, except for Albert's Tile, which is black with a white rune. Birdie's Tile allows her to breathe underwater, while Leroy's Tile gives him a photographic memory.
The second book in the series is The Pillars of Ponderay which has the heroes return to the Core for more adventures.