At a clothing store:
"Hi what's your name?"
"Cool, I am Nate." *handshake*
"Hey I need your opinion on something, do you think clothing A matches clothing B? No? Why not? What do you think would be a nice combination?"
And then you have a variety of ways to close...
At a random line, whether it be comedy, food, anything really...:
"Hi what's your name?"
"Cool, I am Nate." *handshake*
"So what's good here (good days/times to come, good food to order, etc.)"
Then close.
Unless the girl has no class, I am pretty sure she'll tell you her name. Handshakes work exceptionally well because people in the U.S. or any well developed westernized world are conditioned to shake hands when someone puts their hand out.
This is a polite, civilized, and fail proof way to open, what you do afterwards it's up to you
.