Tidus had absolutely no problem with Tifa going first - she'd deck him if he objected. He kind of preferred his jaw in one piece.

The very first thing that Tidus noticed as he stepped away from the entrance was the cold blasting at him, and he shuddered a little. Luca's clothing was not meant for this sort of weather, and he was already starting to freeze. The rocks everywhere also brought back a set of not so happy memories, and he shook his head as he moved with Tifa. "Like Gagazet all over again... Hey! Firion!" he called out, cupping his hands around his mouth to help the sound carry. "Hello! Rosebud?! Where are you!"

No, the nickname would never, ever die.