Q&A for How to Get Shiny Pokémon in Pokémon X and Y

If the Pokemon you're looking for can be found in tall grass, you can try chaining it with the Pokeradar, or, if it lives in water, chain fishing it. Additionally, you might want to do some research and see if it can be found in a Friend Safari, where shinies are eight times more common. Finally, if you can complete the National Pokedex, you'll receive the Shiny Charm, an item which will triple your odds of finding a shiny and stack with any of the other methods described above (except the Friend Safari).

Bulbapedia is probably the best place to go to find what egg group your Pokemon belongs to.

You can only use the Poke Radar in grass or flowers.

Unfortunately, you cannot find the gun fossil in X and Y. In order to obtain a Genesect, you must trade for it or wait for an event where you will be gifted with one. You can also hack for it, but this is not recommended.