Yoshi's Story
Purple Yoshi
Go to any level in story mode and collect all the coins. Once you do this the purple yoshi will appear on the select a yoshi screen. Now you can play as the purple yoshi!
Credit Goes To: Joe Donovan via IGN 64

Black Yoshi(2-2)
Go to Bone Dragons Dungeon in Story mode (page 2, board 2.) Go almost all of the way through the level to the tulips. Go to the tulip to the left of Miss Warp #4. Stand in front of the tulip and eat it. Fire yourself upward and break out of the egg near the ceiling. Pop the "?" bubble and grab the Soccer-Ball like egg then falls out. Now complete the rest of the level as you usually would. When you go to the Character Select Screen, you'll be able to select Black Yoshi.


 Purple Yoshi
 Black Yoshi(2-2)
 White Yoshi(3-2)
 Black Yoshi(2-1)
 Black Yoshi(2-3)
 Black Yoshi(2-4)
 White Yoshi(3-3)
White Yoshi(3-2)
Here's how to find the white Yoshi in Page 3-2 (Tall Tower): Proceed to the first leaf and the message block that introduces it. Drop down to find Miss Warp (2) and a message about a big egg. As you climb the platforms back up, drop to the left and pop the "?" bubble. Grab the white egg, and finish the stage.

Black Yoshi(2-1)
To find the black Yoshi in page 2, level 1, find the tulip that turns you into an egg and shoot yourself up. You will land on a platform with a bubble -- pop it and you get the black Yoshi egg. You will be able to select Black Yoshi once you finish the level. Note that you can only have ONE Black Yoshi at the time.

Black Yoshi(2-3)
Another black Yoshi: Go to Page 2-3. Go through the level until you find a Help Box that tells you about a "big egg."  From there, go to the left and jump on the last floating rock, then jump on the one above that, and shoot the "?" bubble with an egg. You will receive the black egg. Beat the level and you have the black Yoshi. Note that you can only have one black Yoshi at the time.

Black Yoshi(2-4)
Go to the #4 Torrential Maze on Page 2. When you start, keep heading upward through several rooms until you get to a room that has ghost vines coming from the ceiling that you can barely see from the ground. Jump from vine to vine, if you're fast enough, you'll come to an edge with a floating bubble. Pop the bubble and grab the large egg that looks like a soccer-ball. Finish the level as usual by collecting fruit. When you turn to the next page, a black yoshi will be with the other six. This Yoshi likes to eat everything, and he gets three hearts for eating any fruit or shy-guy. His eggs also make a larger explosion than the regular ones.

White Yoshi(3-3)
On Page 3 Area 3 starting from Miss Warp 3 go right. When the road splits with a vase at each end, go in the upper vase. Then go right and enter the upper vase. You should see a dog and a red/orange pipe south of you. Go right and go into the first red/orange pipe to enter a hidden cave. Go right and throw an egg at the bubble. You will now receive a large white egg. Eat the remaining fruit on the level and on the next page and you will be able to choose the White Yoshi.
Credit Goes To: IGN 64