You can get Yuffie in any forest, but Gongaga seems to be the most likely place. If you wanted to get her early, the earliest you can get her is in Junon.

When you do encounter Yuffie in a battle (she's a ninja with huge shuriken), beat her and then approach her after the battle. Don't go for the save point but go straight to her and start a conversation. Reply thusly:
"Not Interested"
"......petrified"
"Wait a second!"
"......that's right"
"......let's hurry on"