Chasing the Golden Rabbit
After collecting 20 stars, a visitor will appear in the basement. A golden rabbit will taunt you while you desperately try to get him. You may be inclined to ignore him, don't! Jump and slide to capture him and, when you do, he'll give you a star. After you collect 50 stars, the rabbit will return to the basement, this time slightly faster. Catch him again, and he's got another star for you.
Owl Rides
In Whomp's Fortress, the fifth star -- "Fall Into The Caged Island" -- is in a cage floating high above Mario. You can spend a frustratingly tedious amount of time trying to get in there with a cannon, but the best way in is to ride Hoot the Owl. Hoot lives in the first tree Mario encounters in Whomp's Fortress. Shake him out by climbing the tree and then jump at him. He'll carry you across the board. Jump again to let go.
Stop The Tick Tock Clock
In the castle's upper floors, you'll find the 14th course, The Tick Tock Clock. Mario must jump into the face of the clock to enter this level. Once inside, you'll see the inner workings of the clock. Spinning gears, sliding belts and swinging pendulums can make traveling through this level a treacherous journey. However, if Mario jumps into the clock just as the big hand is pointing directly at the 12, you can freeze the level.
Secret Visit From Yoshi
After you've collected 120 stars, essentially beating every level, search for the cannon just outside the door of the castle. Jump in and launch yourself toward the castle's roof. Around the corner, you'll find Yoshi. Upon meeting Yoshi, he'll congratulate you on defeating Bowser and award you 100 extra lives plus new jumping powers. You can use these to go back and find anything you missed the first time.