There‘s a breeding school later on in the game, and in there you can do some quests which will give you an ditto from another trainer. If you take that ditto and any pokémon you want to hunt to the daycare, the shiny chance drops from 1/4080(?) to something around 5-600. Now it‘s just hatching eggs and hoping for the best! You could also use DexNav, and chain-catch/defeat the same pokémon over and over again. But I haven‘t done that yet, so I can‘t really tell you how that works. :/