Friends
Friends carry over from game to game (in Gen VI). All you need to do is interact with them and they will re-appear. Note that their Friend Codes must be registered on your device and vice versa.
Acquaintances
These do not carry over. However, interacting with a previous Acquaintance will re-add them to your roster.
Passerby
They will not appear on your new game unless you happen to trade/battle with them again. Period.