Welcome. Now here's how this shit works. First, you need to be in the hall of fame. That'll unlock Kiloude City, which has the FS. To access it, you need friends. Put your friend code in your name and ask for someone to identify your safari. Add them to your friend list on the 3DS main menu. The 3DS friend cap is 100 so deal with that if/when it's an issue. When the friendship is mutual their FS will show up. Only two of the three slots will appear until the friend is recognized as having been entered into the hall of fame and both friends are online and in game at the same time. When your FS is identified, you should place your safari's contents in your name. You can't change your safari without changing your friend code, but I wouldn't worry about it. Someone's gonna want your stuff.

So, why bother with the Friend Safari? First, there are plenty of Pokemon that appear in the friend safari that aren't normally accessible in the cartridge, including all of the non-legendary 6th Gen exclusives. Pokemon found in friend safaris also have an increased rate of being shiny and are guaranteed to have two maximized IVs with potential for more. And to top it off, Pokemon from a Safari with all 3 slots unlocked can be found with their Hidden Abilities.

Possible problem. If you add someone to your friend list before they add you, something weird can occur. If you remain in game, they may appear under your acquaintances. You will be unable to access their benefits until you fix it. Just press home and check your friend list. The DS should verify them then.

Here's a list of every Pokemon that can be encountered in a Friend Safari as well as the same shit I just said.
http://www.serebii.net/xy/friendsafari.shtml

Spreadsheet for tracking which safaris you have and need:
https://www.sendspace.com/file/9nmfk5