In order to get Shiny Pokémon, you need to find, battle, and catch or defeat a species of Pokémon in order to increase the chances of that species being shiny.
According to the respected chart, you start out at 1/4096 chance and work your way up to get the highest chance. With no charm, your highest chances are approx. 1/683 after battling a total of 500 of that species.