The most effective way is to use the Masuda Method with the Shiny Charm. These methods combined bring the Shiny chance up to 1 in 512 (I think?) per egg.
Step 1 will be obtaining the Shiny Charm. In X/Y/OR/AS, that means completing the National Pokedex for everything except the Mythical Pokemon (Mew, Celebi, Jirachi, etc). You will then go to the Game Director and show him your completed Pokedex. He will give you the Shiny Charm as a reward. In Sun/Moon, it is far simpler. You only have to complete the local Alolan Pokedex. This excludes Marshadow and the various Island Scan Pokemon. You then go to the Director, who will then give you the Shiny Charm. In addition to breeding, it also raises the Shiny chance when encountering wild Pokemon, as well.
But this is only half the equation.
Masuda Method makes it even easier to breed Shiny Pokemon. All you need is 2 Pokemon that originated from games played in languages different to each other (not necessarily different from your own). Is there a particular reason you can't put a Ditto on the GTS and ask for a Ditto of a different region? This usually will produce one of a different language, allowing you to use Masuda Method.
Or, you can do like I have done, and go to r/BreedingDittos and jump through the hoops to receive a 6 IV Ditto holding a Destiny Knot. It works quite nicely. Granted, all their Dittos are clones, but that is irrelevant for the Pokemon bred from them.
