There are a few ways to increase your odds of finding Shiny Pokemon in the wild or through breeding. Below are a few methods.
After capturing all 718 Pokemon, you will receive a Key Item from Professor Birch called the Shiny Charm. This item automatically increases your odds of finding a Shiny Pokemon in the wild.
This method was named after Game Freak's Director of the Board, Junichi Masuda.
If you have two Pokemon from different language Pokemon games, you can breed them and raise the odds of finding a shiny Pokemon to 1 in 1366. An example would be breeding a Japanese Charizard with an English Dragonite.
The Pokemon Radar is a Key Item that also helps you find Shiny Pokemon if you use it right. Note: It's not known at this time if the Pokemon Radar will be available in Pokemon Omega Ruby and Alpha Sapphire.
To use this device, go into a large grass patch, activate the Poke Radar, and walk (use the D-Pad, not the Circle Pad). The grass will shake showing where Pokemon are located. There are several differing grass shakes. Remember which type you walked into when the Pokemon Battle started.
Slight Shake: There are no Pokemon here.
Regular Shake: Pokemon will be present here.
Heavy Shake: Pokemon will be present here.
Shining and Shaking: Shiny Pokemon will be present here.
When you first start your Poke Radar, you will have a chance to find a Pokemon. Once you find and defeat that Pokemon, more grass will shake. Grass that is four segments away with the same shake from before will most likely contain the same species of Pokemon you previously fought.
By chaining these Pokemon, you increase your odds of finding a Shiny Pokemon. If you get to 40 encounters of the same Pokemon without breaking the streak, your odds of finding a shiny are 1/50 (Actually 1/200, but 4 grass shakes happen). At this point you do not want to start any new battles, you will want to just recharge the Poke Radar by running back and forth 100 steps without running into any shaking grass.
Here are some more tips on finding Pokemon using this method:
Use Super Repels, this will help you maintain your streak.
Shaking grass that is four segments away will most likely have the same species of Pokemon
Rarer Pokemon are harder to chain.
You can only chain in short grass and flowers.
Quick Balls are your best friend in case you run into a shiny.
Use a tally on your phone or keep a pen and paper to count your chain.
If the music stops and no grass shakes, your chain is broken.
Using a bike, will break your chain.
If no grass shakes, charge the Poke Radar and use it again, sometimes the shaking grass will spawn outside of your area and your chain will continue.
Fishing Rod Chaining
This method uses an old and simple item introduced in the very first generation of Pokémon: the Fishing Rod. All you need to do is use the Old Rod, Good Rod, or Super Rod to catch some Pokemon, and make sure you do not move from your spot. Each time you start a battle from fishing, the chances of getting a Shiny Pokémon increase.
Below are some tips for Fishing Chaining:
It's important to note that the chain will break if you move from the space you are in, so pick one spot and don't move from there. Also, it's better to pick a place surrounded by blocks, rocks, etc., to, corner the Pokémon.
Once you land a Pokémon there's no need to fight it, you can run from the battle and try again. Remember not to move from your spot after the battle.
For better results, get yourself a Pokémon with the ability Suction Cups or Sticky Hold. When a Pokémon with one of these abilities is at the lead of your party, you will always land a Pokemon when fishing. If you don't have one of these abilities, you may get the message "Nothing seems to be biting...", which will break the chain. Also, if you press A too soon or too late, that will cause you to fail landing a Pokémon.