my reference is from gamefaqs, but ill jot down what i know off the top of my head..

the secret dungeon is actually inside the chocobo stables. someone else could prolly elaborate, but in order for the dungeon to "present itself", you have to send three of each level chocobo at the same time to find items in Spira. where you send them depends on what choco you got (timid/normal/bold). so, send three level 1's, 2's, 3's, 4's, and finally have four level 5 chocobos in your active roster (its been a while). i think you send them out at once too.. aaaaanyway once thats done, someone should mention the dungeon.

as for the dungeon itself, ummmmmm... check gamefaqs.com. i think someone posted a map of the dungeon there. other than that, i remember lots of confusion and cursing. good luck