To gain the ability to evolve your pokemon you must beat Rayquazza. After you have beaten Rayquazza and the credits have rolled, you will gain access to Luminous Cave which is found at Whiscash Pond just north of the Pokemon Square. There will be a commotion in Pokemon Square prompting you to go there and learn about evolution. Then the next time you try to leave and go on any mission, your partner will stop you and tell you how to change leaders and travel using just one pokemon. Enter Luminous Cave with just the pokemon that you wish to evolve. The pokemon that you are evolving must meet the required level for evolution and in some cases might need an item such as a moon stone or sun stone. A link cable can also be used. To check if your pokemon is capable of evolution, check it's summary page and under evolution it will be Possible, Not Now, or Not Anymore.