Oval Charm:
Requirement: Beat the pokémon league.
Go to the GAME FREAK office (building to the left of the Dimensional Research Lab up the stairs and to the right). Battle and defeat Morimoto. Morimoto will then give you the Oval Charm.
The Oval Charm increases your chances of receiving an egg from breeding.
Shiny Charm:
Requirement: Catch all the pokémon in the Alola Pokédex
Go to the GAME FREAK office (building to the left of the Dimensional Research Lab up the stairs and to the right). Talk to the director who will give you the Shiny Charm. Note: You may need to talk to him a few times if you haven’t before as he’ll give you stamps for completing each of the island pokédex. Once he’s given you all the stamps he’ll give you the Shiny Charm.
The Shiny Charm increases your chances of finding a shiny pokémon!
Enjoy,
~TheDave