Tutors

In PMU 7, there are many different tutors that teach you moves. These tutors are associated with different items that are required for move relearning and learning. Some range from teaching you the moves that you forgot to teaching you special new moves.

Mushrooms
Mushrooms were introduced in PMU 6. For a brief time, they were removed when transitioning to the new PMU 7. Early in PMU 7, Mushrooms eventually made their return. Mushrooms are the easiest to get out of all the tutor items and are used for the basic move relearning.

The standard Mushrooms—previously named Tiny Mushrooms as to not cause confusion between them and Big Mushrooms—are used at Electivire's move relearn shop (Link Shop) to reteach forgotten moves. (e.g. Ninetales would be able to relearn its level moves). Mushrooms are usually found as a common boss drop, and lower leveled dungeons can have them hidden on the floor. Additionally, players can convert Big Mushrooms into regular Mushrooms via Ripley the Sneasel in Grassroot's Bazaar; slicing one Big Mushroom up into 3 Mushrooms for 500 Poké.

Big Mushrooms
Big Mushrooms are used with "Loomy", a Breloom, and he teaches you a move that your pre-evolved form would have learned (e.g. Ninetales would be able to learn Vulpix's moves) at a certain level. Like Mushrooms, Big Mushrooms can be found hidden in dungeons, or as a drop from bosses.

Shards
Shards are colored fragments of an ancient tool. They were introduced in PMU 6, and for a long period of time, they were void in PMU 7 before finally making a comeback. While they have no use on their own, a Bronzong in Tanren City's bazaar apparently has some use for them, and will teach you Ultra Sun and Ultra Moon BP tutor moves for these shards. Shards can always be found hidden in dungeons, on Floor 21 and higher. There are four different colors of them.

Red Shards
These shards can teach you a random physical move that the Pokémon can be tutored (e.g. Ninetales learning Iron Tail).

Blue Shards
These shards can teach you a random special move that the Pokémon can be tutored (e.g. Ninetales learning Heat Wave).

Green Shards
These shards can teach you a random status move that the Pokémon can be tutored (e.g. Ninetales leaning Role Play).

Yellow Shards
These shards can teach you a random tutor move, which means that it will randomly choose between all three; status, physical, and special. (e.g. Ninetales having a chance at learning Heat Wave, Role Play, or Iron Tail).

Possible Moves
While what moves a Pokémon can get from shards is also dependent on the Pokémon, overall the possible tutor moves are as follows:

Heart Scales
Heart Scales are a valuable item that is desired by a Togetic that lives in Snowbasin Town, Tanren City, and Greenport Town. Togetic will teach you a random egg move out of 5 random ones your Pokémon can learn if you give her a Heart Scale. (e.g. Ninetales learning Tail Slap). They can be found hidden on any tile in dungeons, but most commonly the walls and floor.

Electivire
Electivire can be found in all four regions. Electivire allows the player to remember forgotten moves with a Mushroom.

Breloom
Breloom can be found in 3 of the 4 regions. Breloom teaches your Pokémon previous evolution moves for a Big Mushroom.

Togetic
Togetic can be found in 3 of the 4 regions. Togetic is a tutor that can teach your Pokémon a random egg move for a Heart Scale.

Bronzong
Bronzong is the newest tutor that is released. Bronzong lives in Tanren only and teaches Pokémon a random Ultra Sun and Ultra Moon BP tutor move for a shard that corresponds with the type of the move (Physical, Special, Status).

Trivia

 * Electivire used to teach moves for 500 Poké, later being changed to Mushrooms.
 * Big Mushrooms weren't always used to learn moves prior to evolution.