How to get Vincent in your party

To get Vincent, you must go to the mansion in Nibleheim.  When you go inside, go up the stairs and to your right.  Then search for the room with the safe.  Enter the following numbers:

Right 36

Left 10

Right 59

Right 97

You'll have 20 seconds to open the safe and you must not be a number off.  If you accidentally skip past a number, it won't work.  After you open it, you should come upon a boss.  It isn't that hard so quickly defeat it.  When the battle is over, look on the safe and get the key on it.  Then go to the passage that leads to Sephiroth's library.  Go down the staircase and into the door that you couldn't enter before.  You should see a coffin.  Press O on it and it will open.  You shall see Vincent.  When you talk to him, tell him about your Sephiroth adventure and he will go back to sleep.  Then, open the coffin again and this time ask him about his current situation.  He will tell you and then go back to sleep.  Go back out the tunnel and then Vincent should stop you and tell you that he will join you.

How to get Yuffie in your party

To get Yuffie, you must enter any forest.  Then continue to fight in the forest until you come upon Yuffie.  Once you defeat her, DO NOT go to the save point.  Talk to her and say that you don't want to battle her.  Then say that she is really good and stronger than you.  Then she'll ask you if she could join the group.  Say yes and then your guys will walk away and she will come running after. That is how you get Yuffie.

THINGS YOU CAN DO ONCE YOU GET VINCENT OR YUFFIE

Vincent's Sidequest

Vincent's sidequest allows you to aquire both his level 4 limit break and best weapon.  To get them, you must be in Disc 2.  Go to the middle continent and find a small round lake with a waterfall that is surrounded by mountains.  You must have a gold chocobo, black chocobo, or a green chocobo.  Then, once you have found the area, you should be able to walk on a small ring of grass on the outside edge of the lake.  Mount off your chocobo and go to the left side of the waterfall as if you were facing it.  You should be able to walk into the waterfall.  A short scene will occur and then you will have to go back outside.  Return to the waterfall later on in disc 3 and you can then recieve his level 4 limit break and his best weapon.

Yuffie's Sidequest

After getting Yuffie into your party, go to Wutai (city on the long island on the west side of the map) and then Yuffie will suddenly start talking.  She will then run away.  Go back outside and re-enter Wutai.  You should see a lot of Pagodas everywhere.  You must find Yuffie, so check all the houses and try to find Yuffie.  She will continue to run away until she gets captured.  You are then having to save Yuffie.  Once you have though, you will find that she is a materia stealer.  She will return the materia she stole.  Later you can go to the tall pagoda if you have Yuffie in your party and talk to the guy. He will want to battle Yuffie.  Defeat him and go upstairs and defeat the next person.  Continue doing this until you make it to the top.  You will then recieve the Leviathan materia.