Getting Yuffie in your party requires some time and patience. The earliest you can get her is when you are outside Junon, the town with the big cannon on it, and you can get her in any forest in the game, supposedly. I usually get her in Junon so that I can start her out with my party.
To Get Yuffie:
1. Run around in the forest near Junon until you fight an enemy called "Mystery Ninja" that looks surprisingly like Yuffie. You fight her about one in every ten battles, so be patient -- it may take a while. Also, the enemies can be tough here so get some potions and phoenix downs before you leave Kalm.
2. Fight Yuffie and beat her, instead of the usual screen you will be on a screen with a save point. DO NOT SAVE!
3. Talk to Yuffie. She will act all tough and brash, and challenge you to a rematch. There are 5 times you must respond, and the correct responses, in order, are:
4. You will get to name her, and she will join your party.
If you mess up any of the responses, or use the save point, Yuffie will run away, stealing progressively larger amounts of gil from you, depending on how far along the conversation was. If you can, save yourself the aggravation. When she joins you you get back HALF of what she stole.