There are quite a few methods to Shiny Hunting in this game, most of which have been in use for a number of previous games, but there is one New method that players are using.
The first method is simply by searching in the Wild. This is the worst method by far, only having a rate of about 1 in 8,000.
The second method is by using what is called the Masuda Method, where you put two Pokemon of the same Egg Group in the Daycare that are from different countries in the real world. This greatly improves that odds of hatching a shiny Pokemon to about 6 in 4096.
The third and newest method is hunting for shinies in SOS Battles. This is relatively simple, mainly needing the Wild Pokemon to call for help, having the SOS Pokemon being shiny in odds of about 1 in 500.
Should you obtain the Shiny Charm, this item increases ALL of these rates by 300%, making it much easier to obtain Shiny Pokemon.