The Fairy Ocarina is given to you by Saria the first time you exit Kokiri Forest toward Hyrule Field (across the Lost Woods bridge). This will happen as Child or Adult Link, even if you already have the Ocarina of Time. To skip getting the Fairy Ocarina as a Child, Escape the Forest by hovering to the bridge from the Lost Woods. As long as you do not exit Kokiri Forest through the exit that goes across the bridge, the cutscene will not trigger.
For the rest of the Child portion of the game, you can use Ocarina Items for anything where you would normally use an Ocarina. Once you become an Adult, simply enter Kokiri Forest, then leave through the exit that leads to Hyrule Field. This will trigger the cutscene where Saria gives you the Fairy Ocarina, which will replace the Ocarina of Time if you have it.
(Note: When Adult Link plays the Fairy Ocarina, the Ocarina of Time is always shown in his hands.)