Basically, you just do trade or battle with a passerby, then afterwards he will be an aquaitance. Then, do the one you haven't done yet (battle or trade) and afterwards you have the option to register them as a friend.
I suggest using wonder trade, so you can trade with someone who is playing, they then become your aquaintance, then all you have to do is battle OR trade with them and BAM instant friend!
If you go to the friend safari in Kiloude City, Northeast, then your friend will be randomly assigned to a type of pokemon, to which you can catch three pokemon of that type in there.
Pokemon that are found in the friend safari has:
-A chance to have its hidden ability
-A higher chance of catching (or chaining to) a shiny
-TWO PERFECT IV STATS