In Tandor, there are four Move Tutors willing to teach special moves to your Pokémon. They can be found in the Hotel in Tsukinami Village, in the rightmost room. They each specialize in a style of move: Special attacks, Physical attacks, buffs, and de-buffs. They each require four shards, but each one requires a different color: Special attacks require blue shards, Physical attacks require red shards, Buffs require yellow shards, and Debuffs require green shards.
Additionally, while not exactly tutors themselves, you will find a Move Reminder and a Move Deleter in Venesi City. During a Nuzlocke Challenge the Move Reminder will require a Heart Scale for each move.
Prior to 1.0.3
In Nowtoch City, an old man will teach your Pokémon plenty of status moves for free, but he will only teach moves if you have acquired 4 badges.