There are many ways to hunt for shiny Pokemon. Some ways are faster, or slower, than others, some are harder, or easier.
The fist thing you want to think about is shiny rate, and Pokeballs to catch your shiny in. If you complete the Pokedex, meaning at some point you have obtained every Pokemon, not just having seen it, your professor will give you whats called the shiny charm, which increases the shiny rate to one in every 512 rather than the normal shiny rate of one in every 1/4096. Also the shiny rate nearly doubled in generation six! Awesome!
If you have completed the Pokedex you will have at some point owned every Pokemon. So a great Pokeball to have handy is the Repeat Ball, which has a 4X modifier if you have already caught the Pokemon, which you will have if you go on to get the shiny charm. Another good ball its the Quick Ball which has a 3X modifier when thrown on the first turn of the battle. Dusk Balls are good for hunting at night or in caves, which have a 4X modifier at night or in caves. And of course lastly, the Master Ball which should be saved for a dangerous shiny encounter, like something that can teleport, or self-destruct.
The easiest way to encounter shiny Pokemon is through random encounters. By randomly encountering Pokemon its likely at some point you will find a shiny, but it will take a lot of time. Running around in the grass, fishing, or anyway of encountering Pokemon over and over again is easy, but the rate for finding a shiny is slow.
Another easy way to find shiny Pokemon is Soft Resting or sring. This method can be used by re-encountering the Pokemon over and over until its shiny. This is used by getting a shiny starter or legendary, you encounter the Pokemon, if its not shiny, reset, if it is shiny than catch it. This is good only for starter Pokemon or other Pokemon like legendaries which are usually in a room or a cave where you can safely save before encountering and encounter over an over again until shiny. You can also do this with Pokemon gift from people in game. Like the shuckle given to you in heartgold and soulsilver, or in xy the lapras given to you by another trainer who says he can't care for it. This method of shiny hunting is easy, and speed for finding shinies is medium. Sometimes you have really good luck, sometimes you have really bad luck.
A fast way to get shinies is Horde hunting. In xy you can only use sweet scent or honey in an area where there is no rain, hail, a sandstorm or any weather conditions. In ORAS you can use it no matter the weather conditions. You don’t need to encounter the same horde Pokemon over and over, just keep encountering them, and knock out the horde Pokemon. With the shiny charm the shiny rate is one in ever 512, since you see 5 Pokemon at once it adds up very quickly. I used this method once trying to get a shiny Voltorb in ORAS, I ended up with a Magnemite, but hey its a shiny! I got mine after about an hour without the shiny charm. Be sure you don’t knock out your shiny when you eliminate the other members of the horde!
Another easier option is masuda method breeding. The shiny rate for this method without the shiny charm is one in every 696 or 690 something if I recall correctly and one and every 512 with the shiny charm. This is a easy method and the rate of finding shinies is medium/fast. The way it works is by breeding Pokemon from two different global regions. Like if I wanted a shiny Eevee (which I did shiny breed with this method) I could breed an Eevee from Japan with my American one. This does not mean Pokemon from different in game Pokemon regions. I got my shiny after around 150 eggs without the shiny charm.
A method that requires finishing the game is PokeRadar chaining. I believe the PokeRadar is given to you by the professor or your mother after finishing the game. How it works is you set your radar and walk into a patch of shimmering grass. Once you encounter the Pokemon you want to be shiny knock it out, and walk into fast moving grass to re-encounter it over and over again. You must knock it out each time you encounter it. Once you reach 40 straight encounters, you've maxed your shiny odds. Don't bother walking into grass patches anymore. Once there is a shiny Pokemon the grass will sparkle and have a special sound effect. That means a shiny is in that patch of grass. To reset your Poke radar walk 50 or so steps inside your patch of grass. Remember, you will need a LOT of repels. If you do not use them you risk encountering a Pokemon that's not the kind your looking for and you break your chain. There are a lot of ways to break your chain. One, encountering a Pokemon other than the one your chaining, Two, leaving the patch of grass, Three, quitting the game or saving, Four, walking in the outer edge of the grass. Even if the grass looks good, do NOT go into it. If you go too close to the edge of the grass the game may try to spawn grass where there is none and cause the radar to reset your chain to 0 again. This method is VERY difficult but the rate for finding shinies is fast. You will have to master judging whether the grass that shakes, looks like, and sounds like good grass. Use headphones to better hear the sounds of good grass and pay attention to which grass shakes. If you don't see any good grass, walk around for 50 steps to reset the radar but make sure not to leave the grass, or walk into any of the bad shaking patches.
The last method for xy is chain fishing. Like Poke Radar chaining, you cannot quit the game and must knock out each Pokemon you see. You cannot move from the spot you are in, so repels are not needed. You will want to go to a spot where there are rocks or land around making an inclosed space. Have a Pokemon in the lead spot of your party with the ability suction cups or sticky hold. That will increase the rate of encountering a Pokemon. Unlike chaining with the radar you do not need to re-encounter the same Pokemon over and over again. The shiny rate is increased whether or not you encounter the same Pokemon in a row. Things that can end your chain are moving in ANY way. You cannot move your character. You must stay in the same place. Not pressing A fast enough, or pressing it too early and not encountering a fish will break your chain. The game will either say "you reeled it in too fast" or "you didn't reel it in fast enough" if you make either of those mistakes. Sometimes its not even your fault at all for not landing a Pokemon. The game will say "nothing seems to be biting" that is why you will want a Pokemon with suction cups or sticky hold in the first slot of your party, to decrease the chance of that happening. This method gets shiny Pokemon quickly, but its also very difficult to master if your not the best at fishing, and also there is a limited amount of Pokemon to catch, being as there are only so many aquatic Pokemon you obtain via fishing compared to on land.
I hope this helps
And here is a link to a video that goes into more depth about shiny hunting in all four gen 6 titles
This is where I got all my info and I highly recommend you watch in case I got my numbers wrong. These guys are real experts on the subject I am not. If my link doesn't work for some reason google/youtube Jwittz gen 6 shiny hunting guide the video preview pic should be a shiny Fenekin having just hatched.
I used bulbapedia to double check shiny rates
Also if I have spelling or numerical errors I’m very sorry, trust the video over me, I’m writing this at 1:30 AM