Yuffie's the tricky one. You need a lot of patience to get her, or if you're lucky, this could be one of the easiest parts of the game! Run around in any forested area on the World Map (I always use the ones by Fort Condor) until you encounter a Mystery Ninja. This is where you need the patience, you can go through dozens of random battles before you find her. Once she's found, defeat her, and you'll enter a small cinema screen. DON'T USE THE SAVEPOINT!!! Doing this will make her run away. You'll go through a series of dialogues, in which Yuffie will ask you five questions. You have to answer each one correctly in order to get her to join your party.

Yuffie: You spikey-headed jerk! One more time, let's go one more time!
You answer: Not interested.

Yuffie: You're pretty scared of me, huh?
You answer: ....petrified.

Yuffie: I'm really gonna leave! REALLY!
You answer: Wait a second!

Yuffie: You want me to go with you?
You answer: ....That's right.

Yuffie: All right, I'll go with you!
You answer: .....Let's hurry on.

If you do anything wrong then you will have to start all over again!
 
 

Vincent's the easy one. Basically all you have to do is defeat Lost Number after opening the safe in the Shinra Mansion (It's not that hard!) Once he's taken care of, check the safe again and you'll get a key. In case you need to know, the combination is Right 36, Left 10, Right 59, Right 97. Take the key to the basement down the spiral staircase. In the small room just before entering the library, if you move up you'll see another small room. Go through the door and you'll find Vincent in the coffin. Talk to him about Sephiroth, and when he goes back to sleep wake him again and ask his name. After this he'll refuse to talk to you anymore, so go on into the library. When you are about to leave the mansion, Vincent will meet you and join the team.