I think on the inside I would be freaking out but I would try to keep it cool. Introduce myself, ask for a hug and all that, then go freak out later when she's not around. :)
I would hug her tight and I would be like "OH MY GOD, I LOVE YOU, YOU`RE THE BEST" because, comme on, who wouldn`t say that to her? And then I would probably pass out! *_*
i would give her a hug, take a pic, get an autograph then scream "OH MY GOD IT'S LADY GAGA!!!" and run away while she gets takeled door millions of fans.