There are three great methods, Chain Fishing, Friend Safari, and Masuda Method.
Chain Fishing involves standing in one spot next to a body of water, and fishing repeatedly until you get a shiny Pokemon. It's not a whole lot more complicated than that. The more encounters you get, the higher the odds of getting a shiny gets. If you have the Shiny Charm, the odds can become about 1/100, which is really good! Your chain can break if you move from your single spot that you are standing on, as well as if you miss the Pokemon or one doesn't appear. To make sure one appears, stand near rocks if you can as they make a Pokemon more likely to appear, as well as have a Pokemon with Suction Cups for it's ability if you can.
Friend Safari is a post game area in Kiloude City that allows you go into a field of grass, and two or three Pokemon will appear from the grass, all with a same mutual type. Pokemon will only appear in the field if you have friends on the PSS system. If you don't have any friends on the PSS then you cannot encounter any Pokemon in the Friend Safari. The odds for the Friend Safari with the shiny charm is 1/512.
The final method is the masuda method, which is when you breed two Pokemon from different national origins, ( country ) and you get an egg. That egg has a 1/512 chance of being shiny if you have the Shiny charm. If you breed two Pokemon that are from the same national origin, for comparison, they will have a 1/1365 with shiny charm and 1/4096 without shiny charm.
There is a method with higher odds than masuda method and friend safari, that being the Pokeradar. After 40 encounters in the Pokeradar, your odds of finding a shiny are 1/200, but the Pokeradar is a very difficult and finicky system and I would not recommend using it to get your first few shinies.
If you need more explanation, there are youtube videos out there made to help you.
