First, there are only 4 attacks being taught by the Move Tutors. 3 Old & 1 New. These attacks are also exclusive to a select group of Pokémon, BUT you cannot just offload it on the Pokémon. They need to be at a specific Happiness Level (A High One) Below are the locations and the attacks that are tutored. Click them to go to their attackdex page which contains compatibility lists:

Elemental Hyper Beams

In Route 228, this old man will teach you the elemental hyper beams; Frenzy Plant, Blast Burn & Hydro Cannon.

Move Name	Effect	Type	Kind	Att.	Acc.	PP
Frenzy Plant	User cannot attack on the next turn.			150	90	5
Blast Burn	User cannot attack on the next turn.			150	90	5
Hydro Cannon	User cannot attack on the next turn.			150	90	5

Meteoric Swarm

The attack whose name was given during a competiton is found in a house on Route 210. This old lady shall teach it to any Dragon type Pokémon, and Arceus if its holding the Dragon Plate.

Move Name	Effect	Type	Kind	Att.	Acc.	PP
Draco Meteor	Lowers users SP.ATT two stages			140	90	5