You will eventually encounter Yuffie in a random battle in a forest area. (Any forest area, I recall, though most people get her as soon as they get to the Junon area.) You have to give the correct answers in order to recruit her, the wrong ones will result in Gil being stolen. (I've always gotten them right, so I don't know the severity of having her steal from you.)
Do not use the Save Point or otherwise open up the Camp Menu.
Use these following replies:
"Not Interested"
"...Petrified"
"Wait a second!"
"...That's right."
"...Let's hurry on."
As for getting Vincent, that requires a bit more work. (I don't really care for Vincent in battle, but he is a neat guy I suppose.)
I'd suggest using
this walkthrough on getting Vincent, and possibly everything else in the game.